Chapter 12.3, Problem 1CYP

To determine

Tofind:the probability that Chen draw a king given that he drew a king, queen, or a jack.

Answer to Problem 1CYP

The probability is $\frac{1}{3}$ .

Explanation of Solution

Given:

There is a total $52$ number of cards.

Chen has already drawn a card which is either a king, queen or a jack.

Concept used:

Probability formula:

  $\begin{array}{l}\mathrm{P}\left(\mathrm{A}\right)=\frac{\text{number of favourable events}}{\text{number of total events}}.\\ \mathrm{P}\left(\mathrm{A}\right)=\frac{\mathrm{n}\left(\mathrm{A}\right)}{\mathrm{n}},\mathrm{P}\left(\mathrm{B}\right)=\frac{\mathrm{n}\left(\mathrm{B}\right)}{\mathrm{n}}.\end{array}$

For a conditional probability the formula will be:

  $\mathrm{P}\left(\mathrm{A}|\mathrm{B}\right)=\frac{\mathrm{P}\left(\mathrm{A}\cap \mathrm{B}\right)}{\mathrm{P}\left(\mathrm{B}\right)}$ .

Calculation:

There is a total $52$ number of cards.

Number of kings, queen or jack are $4$ .

The probability of getting a king from the deck is:

  $\mathrm{P}\left(\mathrm{k}\right)=\frac{4}{52}=\frac{1}{13}.$

the probability of having one card that could be either king, queen or jack is:

  $\mathrm{P}\left(\mathrm{J}\mathrm{Q}\mathrm{K}\right)=\frac{12}{52}.$

Conditional probability formula is:

  $\mathrm{P}\left(\frac{\mathrm{K}}{\mathrm{J}\mathrm{Q}\mathrm{K}}\right)=\frac{\frac{1}{13}}{\frac{3}{13}}=\frac{1}{3}.$

Hence, the probability is $\frac{1}{3}$ .