Question

In: Statistics and Probability

A fair coin is tossed until a head appears. Given that the first head appeared on...

A fair coin is tossed until a head appears. Given that the first head appeared on an even-numbered toss, find the probability that it occurred on the second or the fourth toss.

Solutions

Expert Solution

Let A be the probability of flipping heads first after an odd number of flips, and B = (1 - A) the probability that heads is first after an even number of flips.

Then,

Because the first term is the probability 1/2 that heads comes up on the firts flip, and if it doesn't (also with probability 1/2), then you need it to come up after an even number of additional flips, which it will do with probability B.

Substituting B = 1 - A, we get,

Thus, Probability that heads first appears after an even number of flips, B = 1 - A

Now, Probability that first head appears on the second toss =

and, Probability that first head appears on the fourth toss =

Thus,

Probability that first head appears on the second or fourth toss =

Thus, to calculate the Probability that first head appears on the second or fourth toss, given that heads first appears after an even number of flips, we need to use Bayes' Theorem.

So, = Required Probability, and,

= Probability that heads first appears after an even number of flips, given that the first head appears on the second or fourth toss

Thus,

Required Probability = Probability that first head appears on the second or fourth toss, given that heads first appears after an even number of flips =

orchestra answered 1 year ago

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