An unbiased coin tossed 5 times.

Question

An unbiased coin tossed 5 times. $X=k$ when $k$ consecutive heads occur ($k=3,4,5$), else $X=-1$. Find $E(X)$.

Accepted Answer

Counting favourable sequences: $k=3$: 5 ways; $k=4$: 2 ways; $k=5$: 1 way; remaining $32-8=24$ give $X=-1$.

$$E(X) = (-1)\cdot\frac{24}{32}+3\cdot\frac{5}{32}+4\cdot\frac{2}{32}+5\cdot\frac{1}{32} = \frac{-24+15+8+5}{32}=\frac{4}{32}=\frac18$$

Solutions · 1