Question
A fair coin is tossed repeatedly until 5 consecutive heads occur. What is the expected number of coin tosses required to achieve this?
Check your answer
Solution
Answer: 62
Let \(E\) be the expected number of coin tosses required to obtain 5 consecutive heads.
Partition the process according to the first pattern that either succeeds or returns us to the starting state:
\[
E
=
\frac{1}{2}(1+E)
+
\frac{1}{4}(2+E)
+
\frac{1}{8}(3+E)
+
\frac{1}{16}(4+E)
+
\frac{1}{32}(5+E)
+
\frac{1}{32}(5).
\]
These correspond to \(T\), \(HT\), \(HHT\), \(HHHT\), \(HHHHT\), and \(HHHHH\).
Solving:
\[
E=62.
\]
Therefore, the expected number of coin tosses required to obtain 5 consecutive heads is:
\[
62.
\]