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"And" Probability for Dependent Events
Common Core: 7.SP.7a 7.SP.8a S.CP.8

"And" Probability for Dependent Events

Author: Ryan Backman
Description:

Calculate an "and" probability for dependent events.

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Hi. This tutorial covers "and" probability for dependent events. So let's start with a scenario. So let's suppose that 5% of all high school basketball, baseball, and football players go on to play their sports in college. Of these players, 1.7% go on to play professionally. And additionally, 0.01% of high school athletes who do not compete in college will play professionally.

OK, so now let's denote some events that we'll use later. So let event C be the event that a high school athlete will play in college and let event P equal a high school athlete will play professionally. All right, so now let's go ahead and notate each of those probabilities that we just looked at.

OK, so this 5%, again, is 5% of all high school basketball, baseball, and football players go on to play their sports in college. OK, so if we're looking for this probability, using the events that we used before, this 5% will represent the probability of just an athlete playing in college. So the probability of C will equal 0.05. So 5% as a proportion is 0.05.

OK, now, the 1.7%, this one's going to be a little different. So remember that this is of these players. So of the players that play in college, 1.7% will go on to play professionally. So what this probability is is it's a conditional probability. And it's going to be the probability of professional given college. So we're not just interested in all of the high school basketball, baseball, and football players. Now we're looking for just those high school players that played in college. So this probability is 0.017.

Now, the last number-- the 0.01% of high school athletes who do not compete in college will play professionally. So what this is-- again, another conditional probability. But now the condition is that they did not play their sport in college. And this ends up being-- so 0.01% is equivalent to 0.0001. So that represents that probability there.

OK, now let's use a tree diagram to display the sample space of the outcomes of a randomly selected high school athlete. OK, so our outcomes here are first going to be college or not college. So the high school athlete's either going to play in college or not play in college. Then they can either play professionally or not professionally.

OK, now, what's nice about tree diagrams is you can assign probabilities to the branches of the tree that will represent the different outcomes. OK, so we said that there are 5% of athletes-- of high school athletes that go on to play in college. So what I'm going to do is put 0.05 along that branch. There's a 5% chance one of these high school athletes will play their sport in college. And there then is a 1.95 chance that they won't play in college. So these two numbers here need to add to 1, and they do.

OK, now, since playing in college and playing professionally are dependent events-- you're much more likely to play professionally if you played in college than if you're not-- the probabilities that go on these branches need to be conditional probabilities. So the probability that goes here is the probability of pro given college. And we do know that number. That number is 0.017. So there is a 0.017 probability of playing professionally given played in college. So the number that needs to go here then is 0.983. So that number comes from 1 minus 0.017.

Now, the other number we know is the probability of professional given the complement of college-- so playing in the pros if you didn't play in college. And that number was 0.0001. So then what this needs to be is 0.9999. So these two can add up to 1. OK, so that is my tree diagram. And each of the four outcomes are listed.

OK, so now we can-- we'll come back to this tree diagram because this is going to be helpful when we actually calculate probabilities of the four events. But let's start with kind of the formula that we're going to use to calculate these probabilities. All right, so the first rule here, the special multiplication rule, hopefully you've seen before. It says, if events A and B are independent, the probability of A and B equals the probability of A times the probability of B.

OK, so we said that playing in college and playing professionally are not independent. So they're dependent. So we cannot use the special multiplication rule here. So what we need to use instead is what's called the general multiplication rule. Notice that this one doesn't have a condition. This one can be applied to any situation. So this is the probability of A and B equals the probability of A times the probability of B given A.

Now, even if events A and B are independent, the general multiplication rule can still be used. So recall that when events A and B are independent, the probability of B equals the probability of B given A. So whether or not A occurs is irrelevant in independent events because the probability of B given A will simply be the probability of B.

So if we consider the general multiplication rule, the probability of A and B equals the probability of A times the probability of B given A. But if these two are the same, this can simply just be the probability of B. So again, the general multiplication rule can be used whether or not events A and B are dependent or independent.

All right, so let's calculate some probabilities using the general multiplication rule. So let's start with the probability of C and P. We could also use the symbol, the "and" symbol here if we wanted to. So according to the general multiplication rule, probability of A and B is the probability of A times the probability of B given A. So it's B given A.

So if we apply that to this, this is going to be the probability of C times the probability of P given C. So the probability of playing in college times the probability of playing professionally given playing in college. So C and P, that is this outcome here, college and pro. So we're looking at this outcome.

Now, to get that, we said you'd take the probability of playing in college times the probability of playing professional given college. So really, if we just multiplied down the tree, multiply this probability times this probability, that's going to match our formula. So this will end up being 0.05 times 0.017.

And we'll do that in the calculator. And that's going to give me-- so that's scientific notation there. So the e negative 4 means times 10 to the negative fourth. So if I write that out, it's going to be 0.00085. So a pretty small chance that a high school athlete will both play in college and play professionally.

So now let's calculate another probability, the probability of not playing in college and not playing professionally. So again, if we use our formula, it's going to be the probability of not playing in college times the probability of not playing in the pros given not playing in college.

So again, we go back to the tree diagram. This is helpful. We're talking about this outcome here, not playing in college and not playing professionally. So we're, again, going to multiply down the tree because that's going to be the same as applying the formula. So it'll end up being 0.95 times 0.9999. And if we do that-- 0.95 times 0.9999-- sorry, four 9's. And that ends up giving us 0.9499.

So almost a 95% chance that an athlete will not play in college and not play in the pros. So these were a couple examples of applying the general multiplication rule, which, again, helps us calculate "and" probabilities for dependent events. Thanks for watching.

Terms to Know
And Probability for Dependent Events

The probability that two events both occur is the probability of the first event times the conditional probability that the second event occurs, given that the first already has.  Also known as the "General Multiplication Rule)".

Formulas to Know
And Probability for Dependent Events

P left parenthesis A space a n d space B right parenthesis space equals space P left parenthesis A right parenthesis times P left parenthesis B vertical line A right parenthesis