Question

Find the expected number of flips of a coin, which comes up heads with probability p, that are necessary to obtain the pattern h, t, h, h, t, h, t, h.

This is from Sheldon/Ross Introduction to Probability models 11th edition Chapter 3#91. I know there is the textbook solution manual on Chegg, but I am not able to make sense of the solution. I would greatly appreciate if anyone can help me make sense of it!

Answer 1

Answer

Given that the probability of getting on flippig of a coin = p.

Now the given pattern of outcomes of flippig of a coin = h, t, h, h, t, h, t, h

Let us assume it as an event A. That is A= getting the pattern defined above

Then Probability of event A = P(A)= p*(1-p)*p*p*(1-p)*p(1-p)*p , Since the outcomes of coin flipping are independent , where 1-p is probability of getting tail.

So that P(A) = p5(1-p)3.

Now assume that a random variable X following Geometric distribution with event A as its success outcome with probablity of success P(A).

Then pmf of X is

where x refers to number of failures before first success.

Then obviously if X=0, then the success (event A) occurs in first trial , i.e. no failures before first success. That means we get the pattern in the first 8 filips of a coin. Similarly if X=1, then the success(event A) does not occur in first trial but occurs in second trial. That means we get the pattern in 9 flips of a coin.

Then mean of X is given by

Then define Y=X+8; Y gives the numer of flips of a coin to get pattern

. Then expecetd value of Y is

E(Y) gives the expected number of flips of coin to get the pattern.