Question

Two coins are available, one fair and the other two-headed. Choose a coin and toss it once ; assume that the unbiased coin is chosen with probability $\frac{3}{4\cdot}$ Given that the outcome is head, the probability that the twoheaded coin was chosen is

• A. $\frac{3}{5}$
• B. $\frac{2}{5}$
• C. $\frac{1}{5}$
• D. $\frac{2}{7}$

Answer 1

Answer: (B)

Let $F$ denotes fair coin
$T$ denotes two headed
$H$ denotes head occurs
$\therefore P(F)=\frac{3}{4}, P(T)=1-\frac{3}{4}=\frac{1}{4}$
$\therefore P\left(\frac{T}{H}\right)=\frac{P\left(\frac{H}{T}\right) \cdot P(T)}{P\left(\frac{H}{T}\right) \cdot P(T)+P\left(\frac{H}{F}\right) \cdot P(F)}$
(By Baye's theorem).
$=\frac{1 \cdot \frac{1}{4}}{1 \cdot \frac{1}{4}+\frac{1}{2} \cdot \frac{3}{4}}=\frac{2}{5}$