Introductory Statistics: Probability Basics; Events; Rules of Probability (4.1-4.3)

welcome to Mac TV with Professor V in this video we're going to start probability Concepts chapter 4 and we're going to cover sections 4.1 through 4.3

probability Basics events and some rules of probability so terms to know the probability of an

event is the likelihood of the event occurring you could also think of it as the chance of the event occurring

and a probability experiment is a chance process that leads to well-defined results called outcomes something that could be

replicated usually and the sample space is the set of all possible outcomes of a probability experiment so we're going to look at some examples right now

so this terminology can start to make a little bit more sense to you so say we're considering flipping a coin to be a probability experiment and just check that you know it works with our

definition it's a chance process right you don't know what the outcome is going to be every time you flip a coin and we have well-defined results called outcomes it's clear after you flip a

coin what the result is it's not ambiguous and the sample space which is what we're going to be finding here is the set of all of the possible outcomes

so think when you flip a coin just a Fair coin okay nothing fancy schmancy what are the possible outcomes well either the coin lands heads or tails so

I can use the H to represent heads t for Tails sometimes you'll just see sample spaces listed this way or it is a set so if you want to use set notation

and put braces that's all the more proper go you okay next probability experiment we're going to consider is rolling a die we'll just

roll a regular Fair die six sides and so it could lay on any number one through six and I'll list each of those in my sample space so one comma two

three four five and six all right last experiment we're going to consider is flipping two coins okay

so if you're flipping two points there's actually four possible outcomes because if the first coin lands heads the second one could also but if the first coin lands heads the second one could land

tails and then similarly the first coin could land Tails so does the second or first coin lens tales second coin lands

head so there's four outcomes there I'm gonna list them all here in my sample space heads heads Tails tails heads tails Tails heads

all right good now we're going to consider a standard deck of cards so find the sample space for drawing one

card you're just pick in one card from a standard deck and I've just put down here a little image if you're not familiar with the standard deck of cards there's four suits

so here we've got Spades hearts diamonds clubs they go Ace

two three four all the way up to ten and then Jack queen king I mean sometimes aces are considered high it doesn't really matter for probability purposes you just want to know how many and what you're dealing

with and there's 52 cards in a deck total so notice one two three four different suits one two three four there's 13 across here half of the cards are red

half of the cards are black you're going to need to be familiar just with the basics of a standard deck of cards to answer several of the homework exercises that deal with probability okay if

you're not familiar then just go buy a deck of cards and start playing around this weekend that's part of your assignment so find the sample space I mean I'm not going to sit here and list

out each different card but technically there's 52 different outcomes that comprise the sample space when you're drawing one card from a deck so I'm just gonna say that there's 52 different

outcomes I'm not gonna list them all though okay good next example what is the size so just

the size of the sample space for Rolling two dice and here actually we have a little chart to help organize all of the possible outcomes so here's die one

you can roll one through six and then here's die two again you could roll one through six and here's all the possible outcomes or combinations that could

result from Rolling to dice so notice you could get one and one two and one three and one Etc all the way through six and six

so take a second how many different outcomes are here or comprise the sample space there's 36 aren't there

good and this is a basic probability rule for counting that you'll learn when you're trying to figure out how many combinations to make if there's six

outcomes for one event and then six outcomes for the other event then you just multiply them by each other to get the total number of

combinations or outcomes all right very good let's look at something a little different find the sample space for the gender of children if a family has three

children and we're going to use a tree diagram to organize everything so we want to see what are all the different outcomes that are possible for

a family if they're going to have three children are they going to have boy girl girl boy girl you know it can kind of get overwhelming if you try to just sit there and list them all you can do it I have a pretty good systematic way of

doing it but a tree diagram often helps so this is how you set it up for the first Branch this is like baby number one okay either they're going to have a boy or a

girl and then you're going to make another Branch off each of these for baby number two so baby number two could either be a boy

or a girl same thing down here and then baby number three is gonna have a branch off of each of

these now so here's a boy or a girl a boy or a girl baby number three

a boy or a girl boy or a girl okay so you don't have to put the numbers I'm just showing you what each of the branches on the tree represents okay now

here's how you put the sample space together with this you start at baby number one and then you follow each of the branches all the way to the end okay until you've gone up every branch of

your tree so this one would be boy boy okay boy boy boy

then this next one would be boy boy girl boy boy girl then you would have boy girl boy

let's go to the next one boy girl girl and then I've done all of the ones where they have a boy first so then now we go oops I didn't want to erase girl boy boy

and then girl boy girl okay girl boy oh and I think there's one more yeah girl girl

so this is the sample space okay if you want you could put it in set notation how many outcomes are there one two three four five six seven eight eight outcomes

okay very good we'll refer back to this example later okay so just kind of keep it in mind some more terms

an event when we talk about probability an event consists of a set of outcomes

for a probability experiment in a simple event is an event with only one outcome

while a compound event is an event with two or more outcomes so I actually wanted to give you some examples I stuck them down here so if you flip a

coin okay it could either land heads or tails so that's a simple event

but if you flip two coins we saw the sample space earlier it could

be heads heads Heads Tails Tails heads or tails Tails these are compound events because they have two or more outcomes okay

and for events we can use Venn diagrams to represent um probabilities and outcomes which we're going to get more into probabilities in a minute but right now

if we're just talking about outcomes say you have an event like flip a coin and it lands heads so if you call Landing

heads e then not e would be landing Tails which would be everything outside of the event e sometimes certain events can occur at

the same time in which case there's an overlap and this intersection or overlap is what we refer to as two events A and

B happening simultaneously This Is A and B and that's how you would illustrate it and then if you have two events

and you're interested in when either of them happens or the union A or B then both events in their entirety get shaded together okay the overlap and each of

them separately so there's different interpretations for probability we're going to go through three of them first one is classical probability

so classical probability uses sample spaces which we just found to determine the numerical probability that an event will happen and In classical probability

it is assumed that all outcomes are equally likely to occur that's the big Point here so for example when we flip a coin and you're Computing the probability of it lending heads when you

use classical probability you're assuming you have an equal chance of the coin Landing heads or tails is that true there's some discussion right that the

shape of the coin and the way it is on one of the sides tends to make it land a certain way it's actually not 50 50 but you know we don't get into that when we do classical probability

so the formula when you're Computing the probability of any event the probability of event e is the number of outcomes in e or think of it like the

number of different ways that e can happen divided by the total number of outcomes or the size of the sample space and the textbook that we're using right

now it uses the same formula they just kind of call it something different they say P or probability equals F Over N

okay same thing so f is the frequency of the event e so how many different ways it can happen and then n capital N down here is the total number of

outcomes which is the size of your sample space okay so just so you know when you're doing your homework if you see this F divided by n business that's what they're talking about

and then probabilities I mean we can express them as fractions as dead ends just kind of look at the problem if it already is working with decimals then stay with decimals if they've got

fractions stay with fractions usually let the example set the tone and then if they don't say then you can just do whatever you want and general rounding

rule you want to give decimal answers to about two or three decimal places usually the problem will specify how many okay and then your fractions

definitely need to be in lowest terms meaning reduce them okay reduced fractions only don't give me no two tenths okay that's

one-fifth yes all right good so let's look at some examples now remember that sample space that we had earlier where we made the tree diagram for that family that were

they were having three kids very brave of them okay so we're going to talk about them again right now for a family with three children find

the probability that two are girls and one is a boy okay so let's actually go back and look at that sample space we want the ones

where there's two girls and one boy two girls and one boy so that's here and here right those are all of the

outcomes that resulted in two girls and one boy that's three different outcomes and there were eight total so now let me go back here

so this happened three different ways if you want I'll remind you it was girl girl boy girl boy girl and boy girl girl

and there were eight outcomes total so the probability is three divided by eight okay and then like your book might say

the probability if you want to think of it as F divided by n frequency was three n is eight whatever okay

let's look back at our deck of cards example so just a standard deck 52 cards a card is drawn from a deck find the probability of getting an ace

so how many aces are in a deck of cards there's four yes four out of 52.

and this is not reduced your answer has to be in lowest terms so I can divide numerator and denominator by 4 this is 1 over 13.

all right very good same deck of cards find the probability of getting the two of diamonds so the two of diamonds there's only one of those in an entire deck of cards so that

probability is just going to be 1 over 52.

it's already reduced so we're done good okay let's look at a different example in the game of crafts using two dice

okay a person wins on the first roll if a seven or an 11 is rolled find the probability of winning on the first roll and then is

it likely to win on the first roll we'll talk about that okay so game of craps uses two dice and we looked earlier at the size of the sample space when you roll two dice do

you remember how many outcomes there were there's 36 total outcomes okay and the way to win is if you get a 7 or

an 11. all right well how could you roll

an 11. all right well how could you roll a seven let's think about it you've got two dice right two die um so you could get a three and a four

that would give you a seven or a four and a three what else you could roll a two and a five or a five and a two is there anything else

yeah six and one or one and six okay so there's six ways that you could roll a seven you could also win if you roll an eleven

so how many ways can you roll an eleven just two right five and a six or six and five

okay so there's eight different ways to win and there's 36 total outcomes so that means the probability of winning

is going to be 8 divided by 36.

now we have to reduce this so I can divide numerator and denominator by 4 right divide by 4 divided by 4.

so then we're going to have 2 over 9.

okay so that's answering that first question find the probability of winning on the first roll now they're saying is it likely to win on the first roll anytime you ask that

question likely that means more than a 50 chance right or more than half

so is 2 over 9 2 9 is that bigger than one half no it's not right if if you can't compare your fractions then you can just get a common

denominator okay or just think what's half of 9 4.5 right so 4.5 over 9 would be one half so two is way less than that

so since 2 9 is less than one half then we say no it's not likely to win on the first roll okay that's the reasoning so anytime you're asked you're asked something

likely compare it to 50 or one half if it's more yes it's likely if it's not the new if it's exactly right on the dot 50 then

it's equally likely right not more okay let's move on

four basic probability rules so rule number one the probability of an event e is a number

between and including zero and one so we write that the probability of event e is greater than or equal to zero and

less than or equal to one so what this means for us is that a probability is never negative

and it's never greater than one okay so if someone tells you there is a 200 chance that I will come see you tomorrow

they're lying you guys you can't have probabilities greater than one which equates to 100 percent all right example if you roll a die what

is the probability of getting an odd number so how many odd numbers could you roll one three or five

and we know that there's six outcomes right when you roll a die so three are odd out of six total so this is one half and this is

just to illustrate that this probability here one-half is a number between zero and one it's point five as a decimal if you like that better

and it's not negative not greater than one rule number two of probability if an event cannot occur if it's impossible

then the probability is zero so say I asked you the following question if you roll a die what's the probability of getting a number greater than six

well are there any numbers greater than six on a regular diet we're not getting fancy schmancy no right so the probability of that

happening is zero impossible not gonna happen rule number three if an event is certain

then its probability is one if you know for sure it's gonna happen let's look at an example if you roll a die what is the probability of getting a number less

than 10.

well when you roll a standard die all of the numbers are less than 10. so all six outcomes out of six total will be less than 10 which gives me a

probability of one and then the last probability rule is that the sum so if you add up all of the probabilities of all the outcomes in a

sample space again you get one basically all of the probabilities add up to one hundred percent okay those are just four basic facts we're

going to use them to compute probabilities later on complementary events so terms to know the complement of an event e and notice how we spell it

complement this is an e after the L okay not an I like when you complement someone and you tell them your hair looks lovely today that's a different kind of complement

okay the complement of an event e is the set of outcomes in a sample space that are not included

in the outcomes of event e and the complement of e is denoted e with a little bar on the top of it do you see that so that means the complement of E

so let's look at some examples just so we get an idea what an event is and what its complement would represent okay so find the complement of each event

first one is rolling a die and getting an even number what would be the complement of that event it's almost like the opposite but

I don't want to use that word because that's not precise enough but just think not this thing happening so what's equivalent to not getting an even number

getting an odd number so the complement would be rolling a die foreign and getting

an odd number okay so far so good all right let's look at a couple more actually just one more

selecting a card from a standard deck and getting a heart so if that's the event what's not that happening selecting a card from a

standard deck and not getting a heart instead of saying not getting a heart you could also say getting an ace Club

or diamond okay same thing now an event and a compliment have certain properties when it comes to the rules of probability

so if you look at the probability of an event and you add the probability of its complement this will always equal one or equivalently the probability of an

event is equal to 1 minus the probability of its complement and these are interchangeable okay so let's look at some examples and you probably know this you probably do this

on an everyday basis without realizing that you're using a rule that has to deal with complementary events and their probabilities it's kind of common sense for a lot of examples

so first one in 2004 57.2 percent of all enrolled college students were female

choose one enrolled student at random what is the probability that the student was male okay so 57.2 percent

is probability of event e of a female being enrolled in college I'm going to change that to a decimal so 0.572 that's the probability of E and

they want the probability that a randomly selected student is male so that's the complement so probability of

e complement would be 1 minus 0.572 which gives me 0.428 and since the problem gave me the

probability as a percentage originally then I'm just going to write my final answer as 42.8 percent okay another way that you could do this so

if you want to just keep everything in percentages instead of doing one minus then do 100 minus right so just multiply everything by a hundred so or we could

have just done 100 percent minus 57.2 percent and then that gives us the 42.8 percent okay good

so this is all having to do with classical probability now we're going to go into empirical probability and empirical probability does not

assume that all outcomes are equally likely instead empirical probability relies on observation to determine the likelihood

of outcomes so this is a more appropriate time to use frequency divided by n because frequency usually

deals with how many times you observe something when you're collecting data um which is why this is like an empirical approach for probability and

so you have a frequency distribution and the probability of an event e being in a given class or belonging to a certain group is as follows the frequency for

the class divided by the total frequencies in the distribution so same same method to compute it's just you're going to rely on observations or data

that's collected instead of just sitting there and theorizing about what's possible you'll see what I mean in just a second okay so first example 20 people were tested

for their blood types the results were as follows so see we have some sort of data that's being given to us when we're doing empirical probability to help us compute these

um probabilities so here's the blood type a b a b or o and the following frequencies find the probability that

a a person has Type O blood okay so type O in this sample of 20 people there were seven people who had type O

so the probability would be just 7 out of 20.

I'm using this data set this is empirical if we were doing classical probability remember classical assumes everything is equally likely so you would just say oh there's four different

blood types if I'm picking someone at random it should be one-fourth that's not appropriate here though because we know that it's not equally likely for someone to have any one of the four

blood types there's a lot more going on and some are more common than others so collecting data and using empirical probability is more appropriate for this case now do I love that there are only 20

people in the sample no but you know we're doing what we can okay Part B find the probability that a person has Type A or type A B blood

okay so eight people had type a two people had type A B so there's eight of these two of these so all together that's ten outcomes

right out of the 20.

which is one half all right good so this is just empirical because you're using collected data from a sample and as a note the law of large numbers

is a theorem that connects the empirical probability of an event to its theoretical probability so basically the larger your

sample is or the more often you conduct a probability experiment as M gets large then the empirical probability What You observe what you collect

of an event gets closer and closer to the theoretical probability like if you think about it okay I'm going to put it in like a really clear example it is

possible to flip a coin say 10 times and every single time it lands Tails does that mean that oh you have a hundred percent chance when you flip a coin of it Landing tails and zero percent chance

of it Landing heads no that was just a fluke right but the log large numbers tells us well you only flip the coin 10 times if you flip it 100 times or a

thousand times the bigger and bigger the experiment or the more often it's repeated then the likelihood of you getting some fluke lessons and so the

empirical probability What You observe from all your coin flips will approach the theoretical so it should level out and get closer to half and half if you

repeat something often enough which is why we like big sample sizes right when we're collecting data okay enough about that

subjective probability uses a probability value based on an educated guess or estimate and it uses opinions and exact information so there's like no

actual calculation going on it's just you sitting around making up stuff I'm serious so say you apply for a job and

it's you versus one other person okay you and one candidate and in your mind this is why it's subjective in your mind you're like hey they're just as

qualified as I am I think it's a 50 chance that either of us is gonna get the job so that's totally subjective that's just you sitting around thinking you're just

as qualified as them um what's another one you studied all weekend for your Statistics test

and you feel super ready so you're like I got a 99 chance of passing silly stuff like that you'll you'll hear people say things like this all the time now that I've alerted you to it you're

gonna notice it more perhaps it will bother you okay enough about that let's move on to the addition rules for probability

so you're going to hear this term a lot mutually exclusive two events are mutually exclusive if they cannot occur

or cannot happen at the same time so look here events A and B are Illustrated

and notice there's no overlap so A and B these are mutually exclusive they cannot happen at the same time

in the next diagram notice there's this little sliver of overlap right here for events A and B so these two events are

not mutually exclusive why does this matter because it affects how we have to compute the probabilities okay of these events occurring

so let's just first practice identifying whether or not two events are mutually exclusive okay so example one

rolling a die and getting an even number that's one event let's call it event a or roll a die and get a number less than three

so are these two events mutually exclusive can they happen at the same time or not easiest way is just list out all of the ways each of the events can happen list their sample spaces

so for event a if you roll a die and get an even number that could happen if you roll a 2

or a 4 or a six yes how can event B happen you roll a die and you get a number less than three so that happens if you roll a one or a

two I can't include three in there because they said less than three okay now look at these two sets here A and B

is there any overlap is it possible for something to happen that belongs to both A and B at the same time

yes look two two belongs to both sets so are the two events mutually exclusive no they're not mutually exclusive two is

in both they can happen at the same time all right let's look at another example can you draw a card from a standard deck and get a heart or get a spade

well those are two completely different suits you'll never have a card that's a heart and a spade at the same time I'm not going to list out each of the sample spaces there's 13 Hearts there's

13 Spades but none of them overlap so if there's no overlap then we say yes the events are mutually exclusive okay no overlap means yes mutually exclusive

all right so here's our first probability rule to compute when you do have two events that are mutually exclusive the probability that

A or B will occur is the probability of a plus the probability of B okay and if you want let's make a little

diagram here so remember the probability of all of the outcomes in a sample space is always one and here I'm going to have like event a

and we just said that it's mutually exclusive with event B so there's no overlap so if you want the probability of A or B

then you take the probability of a and you add it to the probability of B so you're going to take this plus this

okay good here's an example at a community swimming pool there are two managers eight lifeguards three

concession stand clerks and two maintenance people if a person is selected at random find the probability that the person is

either a lifeguard or a manager okay so all together how many people are working at this swimming pool

two plus eight plus three plus two so there's 15 people total and they want the probability that the person is either a lifeguard or a

manager and you can't have more than one job at this pool okay so there's eight lifeguards and two managers

so we're just gonna add their probabilities together so probability of someone being a lifeguard picked at random would be eight over fifteen probability that someone's a manager is

2 over 15.

and then now we're going to add these together so 8 plus 2 that's 10.

over 15 and reduce your fraction I can divide by 5 right so that's two-thirds good

so good another example at a used book sale a hundred books are adult books and 160

are children's books of the adult books 70 are non-fiction well 60 of the children's books are

non-fiction if a book is selected at random find the probability that it is an adult book

or a children's non-fiction book okay so all together how many books are there at this book sale well there's a hundred

adult books and 160 children's books so 260.

and they want the probability that we get an adult book or children's non-fiction so that's going to be the

probability of getting an adult book plus the probability of a children's non-fiction book

so all together there are a hundred books that are for adults so that's 100 out of 260.

and then how many children's non-fiction are there only 60. so 60 out of the 260. and then

only 60. so 60 out of the 260. and then

add these up so 100 plus 60 that's 160.

don't change the denominator when you add your fractions okay and then look you don't need a calculator we can reduce this on our own the zeros cancel

16 divided by 26 you can cut each of those in half that's 8 over 13.

all right good now you might be wondering well what if the events are mutually exclusive

are not excuse me what if the events are not mutually exclusive well then we have to be careful that we don't count that overlap twice so when you have events that are not mutually exclusive meaning

they can happen at the same time then the probability of A or B it's still probability of a plus probability of B but then you subtract the probability of A and B

watch using the Venn diagram if you take the probability of a you're taking all of this right that I'm shading here

and then if you add to that the probability of B then you're adding this piece

but notice I counted the overlap twice right once with a once with B so this right here that overlap that's the

probability of A and B and that's why we subtract it at the end because it got counted doubly I know okay so that's all you have to do when you have two events that are not

mutually exclusive so let's try let's try here a single card is drawn from a standard deck find the probability of selecting

the following so first probability of getting a 4 or a diamond so that's going to be the probability of getting a four

or I'm going to put plus the probability of getting a diamond and then I'm going to subtract the probability of getting 4 and a diamond

so let's see what's the probability of drawing a 4 in a deck of cards how many are there there's four right out of 52.

and then how many diamonds are in a deck of card cards there's 13.

minus can you get a card that's a four and a diamond you sure can that Four Diamonds right so minus one out of 52.

because if you think about it we counted that four of diamonds twice I counted it here with all the fours and then I counted it again with all the diamonds so we got to take one of them out

4 plus 13 that's 17 minus 1 that gives us 16 out of 52 and then I can divide top and bottom by 4 so this is 4 over 13.

good next example what's the probability of getting a Jack or a black card

okay so that's going to be the probability of getting a jack plus the probability of getting a black card minus the probability of getting a

blackjack so Jack and black all right so how many Jacks are in the deck

there's four right so four out of 52.

plus how many Block cards are on the deck half of them right so 26 out of 52.

minus how many Jacks are black there's two right you have your Spades and your clubs so Jack of Spades Jack of

clubs those are both black and I've doubly counted them so minus two out of 52. so all together we've got 4 plus 26

52. so all together we've got 4 plus 26 that's 30 minus 2 28 out of 52. can we

reduce this guy yep divide by 4 so 7 over 13.

all right which event is more likely to occur so which one has the higher probability this one right here 7 out of 13 is more

than 4 out of 13.

so Jack or black it's more likely okay one more example a local postal carrier distributes

first-class letters advertisements and magazines for a certain date she distributed the following numbers of each type of item the information has been organized into the contingency

table below so she delivers mail to homes and businesses first class letters ads and magazines

first part says if an item of mail is selected at random find these probabilities that the item went to a home so really what I need to do first is get totals so I know how many pieces

of mail we're working with all together so let's start adding everything up total sum up each of the rows each of the columns okay

so how many first class letters were there 1057 how many ads 1427 and magazines 300.

if I add this whole row that'll tell me how much how many pieces of mail she delivered all together and this is 2784.

and then I'm also going to sum up across each of the rows here so all of the home deliveries she delivered 934 pieces in the mail

and the businesses 1850.

and also you guys the 934 and the 1850 right here these should sum up to the total as well so you can double check that you did all your addition correctly that way

so now we have everything we need let's see the first part says if a an item of mail is selected at random find the probability that it went to a home

okay so again we're just gonna do probability is your book loves writing F divided by n so how many home pieces of

mail were delivered there were 934 right we don't care if it was first class ad magazine so 934 divided by the total

27.84 . here's a good example like I'm not

. here's a good example like I'm not going to leave this as a fraction because it's not a simple one where I can look at it and say oh that's about half or less than half you know it's

just like a weird ugly numbers so give it as a decimal punch it in your calculator let's round to three decimal places 0.335 then it actually means something to me I

can look at it and say oh it's about a third of the mail delivered went to a home you know okay next one the item was an ad or

it went to a business four so we need the probability that it was an ad plus the probability that it went to a business

minus the probability it was an ad and went to a business okay so how many were ads 14 27.

out of 27.84 how many went to a business 1850 over 27.84

minus what did we double count how many ads went to businesses 1021. so subtract that we counted those

1021. so subtract that we counted those twice out of 27.84.

okay make it a decimal at the very end don't get don't get don't get out of control 14 27 plus 1850 minus 10 21 that's 22

56.

over 27.84 now can you hear me punching it in the calculator okay I'm as a decimal that's 0.810 good

yes perfect all right good job how do you guys feel you're gonna have some like this in your homework but you should be under control and then I'm just going to do one fancy little example but don't stress okay so

this is just for fun the last idea that we're going to work on in this video is the addition rule for three or more events now if the

events are mutually exclusive then it's pretty easy to compute the probability the probability of a or b or c happening

would just be the probability of a plus the probability of B plus the probability of C but what if the events are not mutually

exclusive meaning some of them can happen at the same time well then the formula gets a little more complicated and there's a Venn diagram here below to help us organize exactly what's going on so let's go through it

together the probability of a or b or c it starts off the same probability of a plus probability of B plus probability of C

then you want to subtract the overlaps where you would be double counting some of the probabilities so then we subtract the probability of A and B that would be

all of this right here and then we subtract the probability of A and C which would be this entire region right here

and then lastly we subtract probability of B and C so then we subtract this portion now

we have to add back the probability of A and B and C which is this you know what let me color it in Black because I don't think you can see it with the

highlighter we're adding back where all three overlap because it got subtracted too many times is the idea okay so

that's the formula you will not be required to do probability for three events on the exam or anything like that but I just wanted to work through one

example with you so you could see how it unfurls okay so there's the Venn diagram let's just go back to a nice card example since hopefully we're all

pretty familiar with that so a single card is drawn from a deck standard deck find the probability of selecting

uh part one says an ace or a 10 or a club all right I don't really feel like writing out Ace 10 clubs so I'm just going to call each of these events

something I'll say let event a b that you draw an ace let event B that you draw a ten

and let event C be that you draw a club all right so what we're trying to do is figure out the probability of a or B

or C happening let's write the formula out so this is probability of a plus probability of B

plus probability of C minus probability of a and B

minus probability of a and C minus probability of B

and c and then we add back probability of a and B that's my and sign I'm running out of space bear with me and C

okay let's go through step by step we're just going to think of our standard deck probability of a what's the probability that you draw an ace out of a standard deck

well how many aces are there there's four right so this is four out of 52.

plus probability of B what's the probability that you draw 10. so how

many tens are in a deck there's four also plus probability of C probability that you draw a club well that's one of the four suits so there's 13 oh why did I

put ten that should be a 52.

there's 13 out of 52.

minus okay let's be careful here probability of A and B so what's the probability that you get an Ace and a 10.

well that's impossible you can't get a card that's simultaneously an Ace and a 10. so the probability is zero out of 52

10. so the probability is zero out of 52 or you could just put zero minus probability of A and C so you get an ace

and at the same time it's a club can that happen why yes if you get the Ace of clubs so one out of 52 for that guy

minus probability of B and C so what's how many tens are there that are also clubs can that happen yes again the Ten of clubs minus one out

of 52.

so here probability of A and B 0. probability of a and c 1 out of 52.

0. probability of a and c 1 out of 52.

probability of b and c 1 out of 52.

who's left probability of A and B and C so can you get a card that is an Ace and a 10 and a club

no way so plus 0 out of 52.

highlight that in yellow and then this is also why I recommend like don't reduce the fractions um until the very end because see like 13 out of 52 is 1 4. but see how lovely

it is right now everything has a common denominator aren't we appreciating okay so four plus four eight

plus thirteen that's 21 minus 1 minus one so 21 minus two that's 19 out of 52.

does that reduce No 19 is prime so we're just gonna box it and be done [Music]

and then example two says what's the probability of a Jack or king or club or heart oh my goodness that's four events

I only gave you a formula for three so that's a challenge problem you can try it on your own if you're feeling adventurous comment down below if you think you have

the answer maybe I'll post it on Instagram and you can Google the formula or see if you know what would be super fun draw out a Venn diagram for four events and

see if you could you know come up with what the formula would be that would be ever so clever but anyways this was all optional just for the exam you're going to be responsible for addition rule with

just two events like we did earlier in the video okay don't stress so please give this video a thumbs up if you enjoyed it subscribe to my channel if you haven't already and then feel free to browse around check out the other

playlists I have video lectures on calc123 obviously the rest of introductory statistics trig pre-calc all that good

stuff so see you guys soon take care bye