Question

An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered 2, 3, 4,..,12 is picked and the number on the card is noted. The probability that the noted number is either 7 or 8, is

• A. $0.24$
• B. $0.244$
• C. $0.024$
• D. None of these.

Answer 1

The correct option is B

$0.244$

Explanation for the correct option:

Step:1 Find out the probability of the number to be $7$.

The probability that the result after tossing the coin is head is $=\frac{1}{2}$.

The probability that the card from the deck is picked is any number between $2$ to $12$ is $=\frac{1}{11}$

The probability that the sum of the numbers of two dices when rolled is $7$

$$\begin{gathered} =\frac{6}{36} \\ =\frac{1}{6} \end{gathered}$$

Therefore, the probability that the noted number is $7$

$$\begin{gathered} =\left(\frac{1}{2}\right)\times \left(\frac{1}{11}\right)+\left(\frac{1}{2}\right)\times \left(\frac{1}{6}\right) \\ =0.129 \end{gathered}$$

Step:2 Find out the probability of the number to be $8$.

The probability that the result after tossing the coin is tail is $=\frac{1}{2}$.

The probability that the card from the deck is picked is any number between $2$ to $12$ is $=\frac{1}{11}$

The probability that the sum of the numbers of two dices when rolled is $7$

$$\begin{gathered} =\frac{6}{36} \\ =\frac{1}{6} \end{gathered}$$

The probability that the sum of the numbers of two dices when rolled is $8$

$=\frac{5}{36}$

Therefore, the probability that the noted number is $8$

$$\begin{gathered} =\left(\frac{1}{2}\right)\times \left(\frac{1}{11}\right)+\left(\frac{1}{2}\right)\times \left(\frac{5}{36}\right) \\ =0.115 \end{gathered}$$

The probability that the noted number is either $7$ or $8$ is

$$\begin{gathered} =0.129+0.115 \\ =0.244 \end{gathered}$$

Therefore, Option(B) is the correct answer.