Question

Two coins are tossed. What is the conditional probability that two heads result, given that there is at least one head ?

• A. $\frac{1}{4}$
• B. $\frac{2}{3}$
• C. $\frac{1}{3}$
• D. $\frac{3}{4}$

Answer 1

Answer: (C)

$\frac{1}{3}$

Let A be the event that two heads result and B the event that there is at least one head.
If S denote the sample space, then $S=\{(H,H),(H,T)(T,H)(T,T)\}$
$A=\left\{(H,H)\right\}$
$B=\{(H,H),(H,T)(T,H)\}$
So, $A\cap B=\{H,H\}$
$P\left(B\right)=\frac{3}{4},$
$P(A\cap B)=\frac{1}{4}$
Hence $P\left(A|B\right)=\frac{P(A\cap B)}{P\left(B\right)}$
$=\frac{1/4}{3/4}=\frac{1}{3}$