Question

An unbiased coin is tossed. If the outcome is head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well shuffled pack of nine cards numbered $1,2,\cdots 9$ is randomly picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$ is :

• A. $\frac{19}{36}$
• B. $\frac{19}{72}$
• C. $\frac{15}{72}$
• D. $\frac{13}{36}$

Answer 1

The correct option is B $\frac{19}{72}$

$P\left(H\right)=\frac{1}{2},P\left(T\right)=\frac{1}{2}$
If outcome is head,
$P\left(7\text{or}8\text{is the sum of two dice}\right)=\frac{11}{36}$
If outcome is tail,
$P\left(7\text{or}8\text{is the number on card}\right)=\frac{2}{9}$
Then the probability that the noted number is either $7$ or $8$ $=\frac{1}{2}\times \frac{11}{36}+\frac{1}{2}\times \frac{2}{9}$
$=\frac{19}{72}$