A coin is biased so head is 3 times as likely as tail.

Question

A coin is biased so head is 3 times as likely as tail. Tossed until a head or 3 tails occur. If $X$ = number of tosses, find the mean of $X$.

Accepted Answer

$P(H)=\dfrac34,\ P(T)=\dfrac14$.

$$E(X) = 1\cdot\frac34+2\cdot\frac34\cdot\frac14\ (\text{TH})+3\cdot\left[\left(\frac14\right)^2\frac34+\left(\frac14\right)^3\right]$$

$$=\frac34+\frac38+3\cdot\frac{4}{64}=\frac34+\frac38+\frac{3}{16}=\frac{21}{16}$$

Solutions · 1