Unseen Coins A statistics professor tosses two coins that cann...

Question

Unseen Coins A statistics professor tosses two coins that cannot be seen by any students. One student asks this question: “Did one of the coins turn up heads?” Given that the professor’s response is “yes,” find the probability that both coins turned up heads.

Accepted Answer

Welcome back, everyone. In a lot to draw, 2 numbers are selected at random from a set of numbers 1 through 10. A spectator asks, is at least one of the numbers added? Given that the answer is yes, what is the probability that both numbers selected are added? Now if we're going to find a probability that both numbers selected are odd, given that at least one of the numbers is odd, then we can use the concept of conditional probability. Let's let P off A for the event A be the probability. That Both numbers are odd, OK, probability that both numbers are odd. And pay off B. With the probability that the probability. That at least one number is odd. OK. List one number is added. Then that would imply the probability of a given b, that is the probability both numbers are odd, given that at least 1 is odd, is equal to the probability of a intersect B. In other words, the probability that both events A and B occur divided by the probability of B. So, if we're going to figure this out, we'll need to find the probability of A and then the probability of A and sorry, the probability of B, and then the probability of A intersect with B, OK? Now let's start with the probability of A. Before we can find any probabilities, we need to know how many pos possibilities we have here, OK? In other words, what are the total number of ways to select two numbers from 10 without regard to order? Well, we can figure that out by using the combination formula. OK? So to select, OK. 2 numbers. From 10 Without regard for order or without regard to order. OK. Then we're going to be finding the combination. Of 10 and 2, OK, where 10 is the total number of items and R is the number of items to choose. And by the combination formula, this is going to be equal to 10 factorial divided by 2 factorial multiplied by 10 minus 2 factorial. This is 10 factorial, OK, divided by 2 factorial multiplied by 8 factorial, and that would be equal to 45, OK? So in other words, there are 45 possibilities here if we're just selecting any two numbers from 10. Now, what's the probability that we can select, or sorry, how many ways can we select two odd numbers from the 5 odd numbers? Well, because remember in this list, if it's 10 numbers. OK. From 1 through 10, then the odd numbers would be 1357, and the 9. So again, we can use the combination formula. So now, to select, to select 2 odd numbers, OK, from the 5 odd numbers. Yeah Then again, by the combination formula, we're finding the combination of 5 and 2, so that's 5 factorial divided by 2 factorial multiplied by 5 minus 2 factorial. Which is 5 factorial divided by 2 factorial, 3 factorial, which is going to be equal to 10. So there are 10 ways, 10 possibilities if we were to select 2 odd numbers from the 5 odd numbers. So now the probability that both numbers are odd. OK, the probability of A is going to be equal to the number of ways to pick two odd numbers divided by the total number of ways to pick any two numbers. So that's 10 divided by 45, which equals 29. And since both numbers are odd, then we can say that. The probability of A intersected with B is going to be equal to the probability of A, OK? So we can keep that in mind because that's going to change our formula a bit, OK? No, we need to find the probability of B, that is at least one number is odd. So how can we do that? Well, it might be easier to find that by calculating or by finding the probability of the complementary event that is both numbers are even and subtracting it from one. Like the probability, both numbers are odd. In our data set, we have 5 even numbers, 2468, and 10. So, the number of ways to pick two even numbers, OK. So to select two even numbers from the 5 numbers that we, from the 5 even numbers that we have, OK, then that probability, or sorry, that combination. Would also be the combination of 5 and 2 which we've already worked out to be 10, OK? Therefore, the probability that both are even. OK, will be equal to 10 divided by 45, which equals 29th, which implies then that the probability both are odd or the probability of B will be equal to 1 minus the probability both are even. That is 1 minus 2/9 which equals 79. Now, let's do a quick check. We have the probability of B 79. We have the probability of a 2/9 and we know it's equal to the probability of A intersected with B. So now, if we put all of that together, this means the probability of a given B, OK, is going to be equal to 2/9, OK, divided by 79, which equals 2/7. Thus, the probability that both numbers selected are odd, given that at least one of the numbers is odd is 2/7. Thanks a lot for watching everyone. I hope this video helped.

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