Chapter 11.4, Problem 1LC

Solution

To find the probability for the given condition.

  $P\left(\text{club|black}\right)=0.50$

Given information:

Card drawn is black.

Concept:

Conditional Probability:

For any two events $A$ and $B$ with $P\left(A\right)\ne 0$ ,

  $P\left(B|A\right)=\frac{P\left(A\text{ and }B\right)}{P\left(A\right)}$

Calculation:

From the definition of conditional probability,

  $P\left(\text{club|black}\right)=\frac{P\left(\text{club and black}\right)}{P\left(\text{black}\right)}$

Now,

  $P\left(\text{club and black}\right)=\frac{13}{52}=0.25$

  $P\left(\text{black}\right)=\frac{26}{52}=0.50$

Put values,

  $\begin{array}{l}P\left(\text{club|black}\right)=\frac{0.25}{0.50}\\ P\left(\text{club|black}\right)=0.50\end{array}$

Conclusion:

Hence, $P\left(\text{club|black}\right)=0.50$ .