00:01
All right, in this problem, we suppose that we have 10 coins such that if the ith coin is flipped, heads will appear with the probability of i over 10, where i is 1 through 10.
00:14
So we have 10 coins.
00:16
When one of the coins is randomly selected and flipped, it shows heads.
00:19
What is the conditional probability that it is the fifth coin? so first, let's remember our formula for conditional probability of a, if an egg, given event b, and that's the probability of a and b over the probability of b.
00:48
And our pdf is going to be i over 10, or i is one through.
01:07
All right.
01:08
So when we rewrite this, we have the probability that it was a five, given that it was a heads in the sample space which is equal to the probability that it was a five and head over the probability that it was a head also recall that there is a one in ten chance that you choose any coin as stated in the problem so let's start writing this probability so for the top part probably that's 5 and ahead, because the coin that you choose and whether it's ahead or not are independent, we can just multiply 1 over 10 chance that it's the fifth coin times the probability that the 5th coin is ahead, which is a 5 over 10 probability.
02:31
And for the probability that it's ahead, we have to add the probability that it's the first coin, times the probability that the first coin is ahead, which is 1 over 10, plus the probability that it was the second coin, times the probability that the second coin is ahead, plus all the way to the probability that is the 10th coin times the 100 % probability that the 10th coin is ahead.
03:17
And now we can rewrite the top as 5 over 100...