Probability
Biased Coin — Mean of Stopping Time
nta_pyq_2023_apr
Grade 12
Question:
A coin with $P(H)=\frac{3}{4}$ is tossed until H or 3 T appear. If $X$ = number of tosses, mean of $X$ is
\dfrac{38}{16}
\dfrac{15}{16}
\dfrac{21}{16}
\dfrac{81}{64}
Step-by-Step Solution
Key Concept: Enumerate: $P(X=1)=\frac{3}{4}$; $P(X=2)=\frac{1}{4}\cdot\frac{3}{4}$; $P(X=3)=\frac{1}{16}\cdot\frac{3}{4}+(\frac{1}{4})^3$. Compute $E[X]=\sum xP(X=x)$.
$E[X]=\frac{21}{16}$.
Correct Answer: 3