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 . $ Then the expected value of $ X $, is

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 . $ Then the expected value of $ X $, is, $ \frac{3}{16} $, $ \frac{1}{8} $, $ \frac{1}{16} $, $ \frac{3}{8} $

MARKS App · Accepted Answer

If k represents the number of consecutive heads and P k is the corresponding probability while an unbiased coin is tossed 5 times, then k 0 1 2 3 4 5 P k 1 2 5 12 2 5 11 2 5 5 2 5 2 2 5 1 2 5 Now, the expected value of variable X is - 1 1 2 5 + - 1 12 2 5 + - 1 11 2 5 + 3 5 2 5 + 4 2 2 5 + 5 1 2 5 = 1 2 5 - 1 - 12 - 11 + 15 + 8 + 5 = 28 - 24 2 5 = 4 2 5 = 1 2 3 = 1 8

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

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