0:00

In this video, we will define what we mean by independent events,

Â learn ways of assessing independence, and

Â introduce the multiplication rule for independent events.

Â 0:10

Two processes are said to be independent if knowing the outcome

Â of one provides no useful information about the outcome of the other.

Â For example, knowing that the coin landed on a head

Â on the first toss, does not provide any useful information for

Â determining what the coin will land on in the second toss.

Â The probability of a head or a tail on the second

Â toss is .5, regardless of the outcome of the first toss.

Â Therefore, outcomes of two coin tosses are said to be independent.

Â 0:41

On the other hand, knowing that the first card drawn from a deck

Â is an ace does provide useful in,

Â useful information for calculating the probabilities of

Â outcomes in the second draw.

Â This is for drawing the cards without replacement, in other words

Â not putting the cards back into the deck after we draw them.

Â For example probability of drawing yet another ace is going to be 3 over 51.

Â We have 51 cards left in the deck, and only three of them are aces.

Â While the probability of drawing a jack is going to

Â be 4 over 51, since we all, still have four

Â jacks left in the deck.

Â Therefore, outcomes of two draws from a

Â deck of cards, without replacement are dependent.

Â 1:24

Based on this definition, we can develop a

Â general rule for checking for independence between random processes.

Â If the probability of an event A occurring, given that event B occurred

Â is the same as the probability of event A occurring in the first place, then events

Â A and B are said to be independent.

Â This rule basically says that knowing B tells us nothing about A.

Â Note that we use this vertical line notation to mean given.

Â Meaning the probability of A given B. So let's put that rule to use real quick.

Â 2:00

In 2013 Survey USA interviewed a random sample of 500 North Carolina residents,

Â asking them whether they think widespread gun ownership protects

Â law abiding citizens from crime, or make society more dangerous.

Â 58% of all respondents said it protects citizens.

Â 67% of white respondents, 28% of black respondents, and 64% of

Â Hispanic respondents shared this view. Based on these we want to fill in the

Â blank in the following sentence.

Â Opinion on gun ownership and race ethnicity

Â are most likely, which of the following?

Â Complementary, mutually exclusive, independent, dependent, or disjoint.

Â These should all be terms that you're familiar with by now.

Â Let's take a look at what we're given.

Â We're given that the probability that a

Â randomly chosen resident believes that guns protect citizens

Â is 0.58.

Â We also know that if the resident is white, then this probability is 0.67.

Â Once again, we use this vertical line notation to say the probability

Â that somebody believes that guns protect citizens, given that they're white.

Â And that probability is 0.67.

Â If they're black, the probability is 0.28, and lastly, if the

Â resident is Hispanic, the probability that they

Â believe that guns protect citizens is 0.64.

Â Since the probabilities of thinking that guns

Â protect citizens vary greatly based on the

Â person's race or ethnicity, Opinion on gun

Â ownership and race ethnicity are most likely dependent.

Â So knowing somebody's ethnicity might actually give us useful

Â information about their opinion on guns, and therefore, we are saying

Â that the two variables are most likely dependent on each other.

Â 3:56

We've been using wording like most likely

Â dependent since we're working with sample data.

Â And we're not yet using statistical inference tools

Â that allow us to take the results that

Â we get from my sa, from our sample and expand that to the population at large.

Â 4:13

If we observe a difference between the conditional probabilities that we

Â calculate based on the sample, we say that these data suggest dependence.

Â The next natural step would then be to actual conduct a hypothesis test.

Â To see if what we observe these difference that we observed,

Â could have just happened due to chance or natural random sampling.

Â 4:33

Or, if there's actually a real difference in the population.

Â We've done a little bit of that at

Â the end of the last unit, and we're going to get

Â back to doing that in the next unit as well.

Â But for now we're kind of picking up building blocks to get us there.

Â However, before we get there, we can

Â actually do a little bit of speculating based

Â on the magnitude of the differences that we observe as well as the sample size.

Â For example, if the observed differences between the conditional

Â probabilities, this is kind of like the probabilities we were

Â just looking at.

Â Probability that guns protect citizens, given that somebody's

Â white versus, given that they're black versus given

Â that they're Hispanic, if these conditional probabilities varied

Â greatly, in other words the differences are large.

Â Then there is stronger evidence that the difference is real.

Â That we would see something similar to that, had

Â we had data from the entire population as well.

Â 5:39

Now that we know how to check for independence, let's see what

Â we can do with events once we find out they're, that they're independent.

Â The product rule for independent events says that if A

Â and B are independent, then the probability of A and

Â B happening is simply the product of their probabilities.

Â 5:58

Say you coss, toss a coin twice.

Â What is the probability of getting two tails in a row?

Â Sounds pretty simple eh?

Â The probability of two tails in a row is simply going to be the probability

Â of a tail on the first toss times the probability of a tail on the second toss.

Â We've seen before we've talked about before that coin tosses are independent

Â of each other.

Â Therefore we're ab, we are able to apply this rule that we've just learned.

Â 6:25

Probability of tail on either toss is simply 0.5 or 1 over 2.

Â So the overall probability is going to be a quarter or about 25%.

Â A quick note, this rule isn't really limited to just two events.

Â And it can actually be expanded to as many independent events as you need.

Â So if,

Â instead of doing two coin tosses, we had a hundred of them.

Â We could simply multiply a hundred of the same probabilities together.

Â Generically said, if A1, A2 all the way through

Â Ak are independent, then probability of all of these

Â events happening at once is simply going to be

Â the product of the individual probabilities of the events.

Â 7:25

Assuming that the obesity rates stay constant, what is the

Â probability that two randomly selected West Virginians are both obese?

Â We're given that 33.5 % of West Virginians are obese which

Â we can denote as probability of being obese as 0.335.

Â It's often useful to make lists of the givens and the

Â problem, as we have been doing in the past couple examples.

Â This helps to keep everything neat and organized and then it help, makes it

Â easier for you to refer back to these values when you need them later in

Â your calculations.

Â 8:00

We're told that the two individuals are randomly selected.

Â Which means that they're going to be independent of

Â each other which, with respect to their obesity status.

Â For example, if we pick two people from the same household

Â and one is obese, the other one might be more likely

Â to be obese as well, given that people who live in

Â the same household are more likely to have shared eating and

Â exercising habits.

Â However, since we're randomly selecting these

Â individuals, we can say that they're independent.

Â And since the two are independent, the probability of both of

Â them being obese will simply be the probability, will simply be

Â the probability of the first one being obese times the probability

Â of the second one being obese, each of which is 0.335.

Â Resulting in an 11% chance of two randomly selected West Virginians being obese.

Â This value, 11% of the probability of both of these people being

Â obese, is less than the probability of either of them being obese.

Â Which makes sense.

Â For two reasons.

Â Mathematically speaking, we're multiplying two values between zero and one.

Â So the product

Â will necessarily be a value lower than either one of them.

Â And conceptually we want to find two people

Â that fit a certain criterion, at the same time.

Â Therefore, the likelihood of us getting what we want should be lower than

Â the likelihood of getting just, finding just one person who fits that criterion.

Â Reasoning through the final numerical answer this way is often useful.

Â It helps us, really understand why the formulas that we're using

Â work the way they do without getting in to theoretical proofs.

Â And it's also useful for checking the final numerical answer

Â in the context of the data that you're working with.

Â In other words, it's really a good way to check your work.

Â