Question

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

Short answer

The total expected number of flips needed to obtain the pattern \(h, t, h, h, t, h, t, h\) is \(\frac{6}{p} + \frac{2}{q} - \frac{1}{pq^2}\), where \(p\) is the probability of flipping a head and \(q=1-p\) is the probability of flipping a tail.

Step-by-step solution

Find the expected number of flips to get the first head

The probability of coming up heads with a flip is p. So, the expected number of flips to get the first head will be the sum of all possible numbers of flips weighted by the probability of flipping a head at each trial. As we know the probability of flipping a head is p, we can use the geometric distribution formula, where the expected value is equal to \(\frac{1}{p}\).

Find the expected number of flips to get the first tail after the head

After getting the first head, we need to obtain the first tail. The probability of flipping a tail is denoted by q, where \(q=1-p\). Now, the probability of flipping a tail after the head (given we are in this position) is q. Using the geometric distribution formula again, we find that the expected number of flips needed to get the first tail after the head would be \(\frac{1}{q}\).

Find the expected number of flips needed for the rest of the pattern

We repeat the process by calculating the expected number of flips needed for the next part of the pattern: \(h, h, t\). We first flip a head (probability p) and then another head (probability p), and then flip a tail (probability q). In this case, the expected number of flips needed for obtaining this part of the pattern will be \(\frac{1}{pq^2}\). Continuing the pattern, we need to calculate the expected number of flips needed for the remaining part: \(h, t, h\): -Expected flips to obtain the head: \(\frac{1}{p}\) -Expected flips to obtain the tail: \(\frac{1}{q}\) -Expected flips to obtain the final head: \(\frac{1}{p}\)

Calculate the total expected number of flips needed

Summing up the expected number of flips needed for each part of the pattern will give us the total expected number of flips: Total expected number of flips = \(\frac{1}{p} + \frac{1}{q} + \frac{1}{pq^2} + \frac{1}{p} + \frac{1}{q} + \frac{1}{p}\) = \(\frac{6}{p} + \frac{2}{q} - \frac{1}{pq^2}\)