Question

In a single throw of two dice what is the probability of obtaining a number greater than $7$, if $4$ appears on the first dice

• A. $\frac{1}{3}$
• B. $\frac{1}{2}$
• C. $\frac{1}{12}$
• D. None of These

Answer 1

The correct option is B

$\frac{1}{2}$

Explanation for correct option:

Step 1: Determining the Probability of event A, B and $A\cap B$

If two dice are thrown ,the total number of sample space = $36$

Let A be the event that first die contains $4$, then the sample is $\left\{\left(4,1\right),\left(4,2\right),\left(4,3\right),\left(4,4\right),\left(4,5\right),\left(4,6\right)\right\}$

then , P(A)=$\frac{6}{36}$

Let B be the event that the number greater than $7$,then the sample is $\left\{\left(4,4\right),\left(4,5\right),\left(4,6\right),\left(5,3\right),\left(5,4\right),\left(5,5\right),\left(5,6\right),\left(6,2\right),\left(6,3\right),\left(6,4\right),\left(6,5\right),\left(6,6\right)\right\}$

then, P(B)=$\frac{12}{36}$

Let $A\cap B$ be the event that first die contain $4$ and the number greater than $7$, that is $\left\{\left(4,4\right),\left(4,5\right),\left(4,6\right)\right\}$

then, P($A\cap B$) = $\frac{3}{36}$

Step 2: Apply the concept of conditional probability

Hence,

$$\begin{gathered} P\left(\raisebox{1ex}{$B$}\!\left/ \!\raisebox{-1ex}{$A$}\right.\right)=\raisebox{1ex}{$P\left(A\cap B\right)$}\!\left/ \!\raisebox{-1ex}{$P\left(A\right)$}\right. \\ P\left(\raisebox{1ex}{$B$}\!\left/ \!\raisebox{-1ex}{$A$}\right.\right)=\raisebox{1ex}{$\left({\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$36$}\right.}\right)$}\!\left/ \!\raisebox{-1ex}{$\left({\displaystyle \raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$36$}\right.}\right)$}\right. \\ P\left(\raisebox{1ex}{$B$}\!\left/ \!\raisebox{-1ex}{$A$}\right.\right)=\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$6$}\right. \\ P\left(\raisebox{1ex}{$B$}\!\left/ \!\raisebox{-1ex}{$A$}\right.\right)=\frac{1}{2} \end{gathered}$$

Therefore, the Probability of obtaining a number greater than $7$, if $4$ appears on the first dice is $\frac{1}{2}$

Hence, the correct option is B.