Question

An unbiased coin is tossed $5$ times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k=3,4,5$, otherwise $X$ takes the value $-1$. The expected value of $X$, is

• A. $\frac{3}{16}$
• B. $-\frac{3}{16}$
• C. $\frac{1}{8}$
• D. $-\frac{1}{8}$

Answer 1

The correct option is C $\frac{1}{8}$

$k=$ No. of consecutive heads
$P(k=3)=\frac{5}{32}(HHHTH,HHHTT,THHHT,HTHHH,TTHHH)P(k=4)=\frac{2}{32}(HHHHT,THHHH)P(k=5)=\frac{1}{32}\left(HHHHH\right)P(\overline{3}\cap \overline{4}\cap \overline{5})=1-(\frac{5}{32}+\frac{2}{32}+\frac{1}{32})=\frac{24}{32}$

$\sum XP\left(X\right)=(-1\times \frac{24}{32})+(3\times \frac{5}{32})+(4\times \frac{2}{32})+(5\times \frac{1}{32})=\frac{1}{8}$