Chapter 29.8, Problem 15SRE

(a)

To determine

Obtain the chance of getting only kings in a deck.

Answer to Problem 15SRE

The chance of getting only kings in a deck is 0.000181.

Explanation of Solution

Calculation:

In general, a standard deck of cards contains 52 cards, of which 26 are red and 26 are black, 13 are of each suit (hearts, diamonds, spades, and clubs) and of which 4 are of each denomination (A, 2 to 10, J, Q, and K). There are 4 face cards. That is, jacks J, queen Q, and kings K.

The probability of an event is given below:

$\text{Probability}=\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}$

Here, 4 of 52 cards in the standard deck of cards are kings.

The probability is given below:

$P\left(\text{First card is king}\right)=\frac{\text{4}}{\text{52}}$

Once a king has been selected, 3 of 51 cards in the standard deck of cards are kings.

The probability is given below:

$P\left(\text{Second card is king}|\text{first card is king}\right)=\frac{\text{3}}{\text{51}}$

Once two kings have been selected, 2 of 50 cards in the standard deck of cards are kings.

The probability is given below:

$P\left(\text{Third card is king}|\text{two card is king}\right)=\frac{\text{2}}{\text{50}}$

General multiplication rule:

$P\left(A\cap B\right)=P\left(A\right)\times P\left(B|A\right)=P\left(B\right)P\left(A|B\right)$

Therefore, the chance of getting only kings in a deck is given below:

$\begin{array}{c}P\left(\text{Only kings}\right)=\left\{\begin{array}{l}P\left(\text{First card is king}\right)\times P\left(\begin{array}{l}\text{Second card is king}|\\ \text{First card is king}\end{array}\right)\\ \times P\left(\begin{array}{l}\text{Third card is king}|\\ \text{First two cards is king}\end{array}\right)\end{array}\right\}\\ =\frac{4}{52}\times \frac{3}{51}\times \frac{2}{50}\\ =\frac{24}{132,600}\\ =0.000181\end{array}$

(b)

To determine

Obtain the chance of getting no kings in a deck.

Answer to Problem 15SRE

The chance of getting no kings in a deck is 0.7826.

Explanation of Solution

Calculation:

Here, 48 of 52 cards in the standard deck of cards are not kings.

The probability is given below:

$P\left(\text{First card is not king}\right)=\frac{\text{48}}{\text{52}}$

Once a non-king has been selected, 47 of 51 cards in the standard deck of cards are not kings.

The probability is given below:

$P\left(\begin{array}{l}\text{Second card is not king}|\\ \text{first card is not king}\end{array}\right)=\frac{\text{47}}{\text{51}}$

Once two non-kings have been selected, 46 of 50 cards in the standard deck of cards are not kings.

The probability is given below:

$P\left(\begin{array}{l}\text{Third card is not king}|\\ \text{two card is not king}\end{array}\right)=\frac{\text{46}}{\text{50}}$

Therefore, the chance of getting no kings in a deck is given below:

$\begin{array}{c}P\left(\text{No kings}\right)=\left\{\begin{array}{l}P\left(\text{First card is not king}\right)\times P\left(\begin{array}{l}\text{Second card is not king}|\\ \text{First card is not king}\end{array}\right)\\ \times P\left(\begin{array}{l}\text{Third card is not king}|\\ \text{First two cards is not king}\end{array}\right)\end{array}\right\}\\ =\frac{48}{52}\times \frac{47}{51}\times \frac{46}{50}\\ =\frac{103,776}{132,600}\\ =0.7826\end{array}$

(c)

To determine

Obtain the chance of getting no face cards.

Answer to Problem 15SRE

The chance of getting no face cards in a deck is 0.4471.

Explanation of Solution

Calculation:

Here, 40 of 52 cards in the standard deck of cards are not face cards.

The probability is given below:

$P\left(\text{First card is not face cards}\right)=\frac{\text{40}}{\text{52}}$

Once a non-face card has been selected, 39 of 51 cards in the standard deck of cards are not face cards.

The probability is given below:

$P\left(\begin{array}{l}\text{Second card is not face card}|\\ \text{first card is not face card}\end{array}\right)=\frac{\text{39}}{\text{51}}$

Once two non-face cards have been selected, 38 of 50 cards in the standard deck of cards are not face cards.

The probability is given below:

$P\left(\begin{array}{l}\text{Third card is not face card}|\\ \text{two card is not face card}\end{array}\right)=\frac{\text{38}}{\text{50}}$

Therefore, the chance of getting no face cards in a deck is given below:

$\begin{array}{c}P\left(\text{No face card}\right)=\left\{\begin{array}{l}P\left(\text{First card is not face card}\right)\times P\left(\begin{array}{l}\text{Second card is not card}|\\ \text{First card is not card}\end{array}\right)\\ \times P\left(\begin{array}{l}\text{Third card is not face card}|\\ \text{First two cards is not card}\end{array}\right)\end{array}\right\}\\ =\frac{40}{52}\times \frac{39}{51}\times \frac{38}{50}\\ =\frac{59,280}{132,600}\\ =0.4471\end{array}$

(d)

To determine

Obtain the chance of getting at least one face card.

Answer to Problem 15SRE

The chance of getting at least one face card in a deck is 0.5529.

Explanation of Solution

Calculation:

From Part (d), the chance of getting no face cards in a deck is 0.4471.

Therefore, the chance of getting at least one face card in a deck is given below:

$\begin{array}{c}P\left(\text{At least one face card}\right)=1-P\left(\text{No face card}\right)\\ =1-0.4471\\ =0.5529\end{array}$