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 $\frac{3}{4}$. Given that the outcome is head the probability that the two-headed coin was chosen, is

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

Answer 1

Answer: (B)

$\frac{2}{5}$

Let $F$ denotes fair coin.
$T$ denotes two headed,

$H$ denotes head occurs

$\therefore P\left(F\right)=\frac{3}{4},P\left(T\right)=1-\frac{3}{4}=\frac{1}{4}$

$\therefore P\left(\frac{T}{H}\right)=\frac{P\left(\frac{H}{T}\right)\cdot P\left(T\right)}{P\left(\frac{H}{T}\right)\cdot P\left(T\right)+P\left(\frac{H}{F}\right)\cdot P\left(F\right)}$ (By Baye's theorem)

$=\frac{1\cdot \frac{1}{4}}{1\cdot \frac{1}{4}+\frac{1}{2}\cdot \frac{3}{4}}=\frac{2}{5}$