Question

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

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

Answer 1

The correct option is A

$\frac{1}{8}$

Explanation for the correct option:

Calculate the expected value of $X$.

Here, $k$is the number of times head occurs consecutively.

$k$	0	1	2	3	4	5
$P\left(k\right)$	$\frac{1}{32}$	$\frac{12}{32}$	$\frac{11}{32}$	$\frac{5}{32}$	$\frac{2}{32}$	$\frac{1}{32}$

Therefore, the expected value of $X$is

$$\begin{gathered} \sum _{k=0}^{5}XP\left(k\right) \\ =(-1)\times \frac{1}{32}+(-1)\frac{12}{32}+(-1)\times \frac{11}{32}+3\times \frac{5}{32}+4\times \frac{2}{32}+5\times \frac{1}{32} \\ =\frac{1}{8} \end{gathered}$$

Therefore, Option(A) is the correct answer.