A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If X denotes the number of tosses of the coin, then the mean of X is

Question

A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If X denotes the number of tosses of the coin, then the mean of X is, 38 16, 15 16, 21 16, 81 64

MARKS App · Accepted Answer

It is given that P ( H ) = 3 P ( T ) ∴ P ( H ) = 3 4 , P ( T ) = 1 4 Since coin is tossed till either 1 H or 3 T occurs. So, process will end in the maximum of 3 throws. X i 1 2 3 P X i 3 4 1 4 · 3 4 1 4 × 1 4 × 1 4 + 1 4 × 1 4 × 3 4 Now we know that, Mean is given by ∑ i = 1 n X i P X i Mean = 1 × 3 4 + 2 × 3 16 + 3 × 1 16 = 21 16 Hence, this is the correct option.

$X i$	$1$	$2$	$3$
$P (X i)$	$\frac{3}{4}$	$\frac{1}{4} \cdot \frac{3}{4}$	$\frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} + \frac{1}{4} \times \frac{1}{4} \times \frac{3}{4}$

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