Chapter 4.3, Problem 13E

Drawing a Card Four cards are drawn from a desk without replacement. Find these probabilities.

a. All cards are jacks.

b. All cards are black cards.

c. All cards are hearts.

To determine

To find: The probability that all 4 cards are jacks.

Answer to Problem 13E

The probability that all 4 cards are jacks is $\frac{1}{270,725}$.

Explanation of Solution

Given info:

The number of jacks in a desk is 4.

Calculation:

Here, the selections are made without replacement. Thus, the selections are dependent.

Multiplication rule for dependent events:

If the events A and B are dependent, the probability of occurring of event B is affected by event A, that is $P\left(A\text{ and }B\right)$ is $P\left(A\right)\times P\left(B|A\right)$.

Then the probability that all 4 cards are jacks is,

$\begin{array}{c}P\left(\begin{array}{l}\text{All 4 cards}\\ \text{are jacks}\end{array}\right)=P\left(\begin{array}{l}\text{First card is jack and }\\ \text{Second card is jack given that}\\ \text{first card is jack and}\\ \text{Third card is jack given that}\\ \text{both first and second card is jack and}\\ \text{Fourth card is jack given that}\\ \text{first, second and third card is jack }\end{array}\right)\\ =\left[\begin{array}{l}P\left(\text{First card is jack}\right)\times \\ P\left(\text{Second card was jack | First card is jack }\right)\times \\ P\left(\text{Third card was jack | First and second card is jack}\right)\times \\ P\left(\text{Fourth card was jack | First, second and third cards are jack}\right)\end{array}\right]\\ =\frac{4}{52}\times \frac{3}{51}\times \frac{2}{50}\times \frac{1}{49}\\ =\frac{1}{270,725}\end{array}$

Thus, the probability that all 4 cards are jacks is $\frac{1}{270,725}$.

To determine

To find: The probability that all 4 cards are black cards.

Answer to Problem 13E

The probability that all 4 cards are black cards is $\frac{46}{833}$.

Explanation of Solution

Given info:

The number of black cards in a desk is 26.

Calculation:

Here, the selections are made without replacement. Thus, the selections are dependent.

Then the probability that all 4 cards are black cards is,

$\begin{array}{c}P\left(\begin{array}{l}\text{All 4 cards }\\ \text{are black cards}\end{array}\right)=P\left(\begin{array}{l}\text{First card is black card and }\\ \text{Second card is black cards given that}\\ \text{first card is black cards and}\\ \text{Third card is black cards given that}\\ \text{First and second card are black cards and}\\ \text{Fourth card is black cards given that}\\ \text{first, second and third cards are black cards }\end{array}\right)\\ =\left[\begin{array}{l}P\left(\text{First card is black cards}\right)\times \\ P\left(\text{Second card is black cards | First card was black }\right)\times \\ P\left(\text{Third card was black cards | First and second cards is black}\right)\times \\ P\left(\text{Fourth card was jack | first, second and third cards are black}\right)\end{array}\right]\\ =\frac{26}{52}\times \frac{25}{51}\times \frac{24}{50}\times \frac{23}{49}\\ =\frac{46}{833}\end{array}$

Thus, the probability that all 4 cards are black cards is $\frac{46}{833}$.

To determine

To find: The probability that all 4 cards are hearts.

Answer to Problem 13E

The probability that all 4 cards are hearts is $\frac{11}{4,165}$.

Explanation of Solution

Given info:

The number of hearts in a desk is 13.

Calculation:

Here, the selections are made without replacement. Thus, the selections are dependent.

Then the probability that all 4 cards are hearts is,

$\begin{array}{c}P\left(\begin{array}{l}\text{All 4 cards }\\ \text{are hearts}\end{array}\right)=P\left(\begin{array}{l}\text{First card is heart and }\\ \text{Second card is heart given that}\\ \text{first card is heart and}\\ \text{Third card is heart given that}\\ \text{first and second card is heart and}\\ \text{Fourth card is heart given that}\\ \text{first, second and third cards are heart }\end{array}\right)\\ =\left[\begin{array}{l}P\left(\text{First card was heart}\right)\times \\ P\left(\text{Second card is heart | First card is heart }\right)\times \\ P\left(\text{Third card is heart | First and second card is heart}\right)\times \\ P\left(\text{Fourth card is heart | First, second and third card is heart}\right)\end{array}\right]\\ =\frac{13}{52}\times \frac{12}{51}\times \frac{11}{50}\times \frac{10}{49}\\ =\frac{11}{4,165}\end{array}$

Thus, the probability that all 4 cards are hearts is $\frac{11}{4,165}$.