### Video Transcript

A shelf in a bookstore contains 30 copies of a book, four of which have been signed by the author. A customer takes one of the copies off the shelf at random and looks through it. Some time later, another customer comes along and does the same thing. And then we have four different questions based on this scenario. What is the probability that both customers take a signed copy off the shelf if the first customer buys the copy they look through? What is the probability that both customers take a signed copy off the shelf if the first customer puts their copy back on the shelf after looking through it? What is the probability that neither customer takes a signed copy off the shelf if the first customer buys the copy they look through? And finally, what is the probability that exactly one of the customers takes a signed copy off the shelf if the first customer puts their copy back on the shelf after looking through it?

We’ll take each one of these scenarios in turn. For part one, we want to know what is the probability that both customers take a signed copy off the shelf if the first customer buys the copy they look through. This is dealing with the probability of two different events. We have to determine if these events are dependent or independent. Since the first customer buys the copy that they look through, it will affect the number of books on the shelf. And therefore, it will affect the probability of the second customer taking a signed copy. And that means we can identify these events as dependent events. For dependent events, the probability of 𝐴 and 𝐵 both occurring will be equal to the probability of 𝐴 multiplied by the probability of 𝐵 given 𝐴 has occurred.

For us, the probability 𝐴 would be the probability that customer one takes a signed copy off the shelf. There are four signed copies out of the 30 copies. The probability that customer A takes a signed copy is four out of 30. Now, if customer one buys their book, there will only be 29 books left on the shelf. But also if 𝐴 occurs, then person one has taken home a signed copy. And there would therefore only be three remaining signed copies on the shelf. Three of the remaining copies on the shelf are signed by the author, and there are now 29 copies left. Before we multiply these values together, we can simplify. Both three and 30 are divisible by three, and both four and 10 are divisible by two. Two times one is two, and five times 29 is 145.

The probability then that both customers take a signed copy off the shelf if the first customer buys the copy they look through is two out of 145.

For part two, what is the probability that both customers take a signed copy off the shelf if the first customer puts their copy back on the shelf after looking through it? Again, we have two different events. But since the first customer puts their copy back on the shelf, this time, the events are independent of one another. What the first customer does does not affect the outcome of what the second customer will do. For independent events, the probability of 𝐴 and 𝐵 occurring will be equal to the probability of 𝐴 times the probability of 𝐵. The first event is the probability that the first customer takes a signed copy off the shelf. There are four signed copies and 30 books on the shelf. That probability is four thirtieths.

But since the first customer puts the book back, the probability of the second customer picking up a signed copy is still four thirtieths. To find the probability of 𝐴 and 𝐵, we need to multiply four over 30 times four over 30. Both of these fractions reduce to two fifteenths, and two times two is four. 15 times 15 is 225. The probability that both customers take a signed copy if the first customer puts the copy back on the shelf after looking at it is four out of 225.

For part three, what is the probability that neither customer takes the signed copy off the shelf if the first customer buys the copy they look through? Again, we want to consider what kind of events these are, independent or dependent. But since customer one is going to buy the copy they look through, these will be dependent events. And for dependent events, the probability of 𝐴 and 𝐵 will be equal to the probability of 𝐴 times the probability of 𝐵 given that 𝐴 occurs. This time, we want the probability that neither customer takes a signed copy off the shelf. There are 26 copies that do not have the author signature. The probability that the first customer takes one of those will be 26 out of 30.

Now, we know that the first customer is going to purchase the book they look through. And that means when the second customer comes along, there will only be 29 copies on the shelf. We’re basing this on the fact that customer one has in fact purchased an unsigned copy. We started with 26 unsigned copies, but since customer one bought one, there are now only 25 copies that are not signed. The probability of 𝐴 and 𝐵 here will be equal to 26 over 30 times 25 over 29. 25 over 30 reduces to five over six, and 26 and six are both divisible by two. So in the numerator we’ll have 13 times five and in the denominator three times 29, which gives us 65 over 87. The probability that neither customer takes a signed copy off the shelf if the first customer buys the copy they look through is 65 out of 87.

And the fourth part, what is the probability that exactly one of the customers takes a signed copy off the shelf if the first customer puts their copy back on the shelf after looking through it? We need to think carefully about what’s happening here. We wanna know that the probability that exactly one of the customers takes a signed copy off the shelf. And there are two ways this can happen. That would be customer one selects a signed copy and customer two does not. Or, customer one does not select the signed copy and customer two does. We’re looking for the probability of something or something else.

And so it’s important to say here that when we’re looking at these either/or situations, we have to decide if these events are mutually exclusive or not. And that means, can both of these things be true at the same time? That phrase “exactly one of the customers” clues us in that these two events are mutually exclusive. They can’t both happen at the same time. For mutually exclusive events, the probability of 𝐴 or 𝐵 occurring will be equal to the probability of 𝐴 plus the probability of 𝐵. That means we’ll need to calculate the probability that customer one gets a signed copy and not customer two, then separately calculate the probability that customer one does not get a signed copy but customer two does, and then add those two values together.

Since customer one is putting their copy back on the shelf, these events are independent events. And when we are dealing with independent events, the probability of 𝐴 and 𝐵 will be equal to the probability of 𝐴 times the probability of 𝐵. The probability that customer one gets a signed copy and customer two does not will be equal to four over 30 times 26 over 30. Remember, customer one is putting the book back. So the probability that customer one picked up a signed copy is four out of 30, and the probability that customer two did not pick up a signed copy is 26 out of 30. We can reduce these fractions to two over 15 and 13 over 15, which when we multiply together gives us 26 over 225.

The second probability would be the probability that customer one does not get a signed copy but customer two does. The chances of customer one picking up an unsigned copy are 26 out of 30. Then customer one puts that book back. And the chances of customer two picking up a signed copy would then be four out of 30. Again, we’ll have 13 over 15 times two over 15, which is 26 over 225. Now, the probability that either one of these things will happen will be their combined probability, 26 over 225 plus 26 over 225, which tells us that the probability exactly one of the customers takes the signed copy must be equal to 52 out of 225.