Chapter 4.4, Problem 40E

Blackjack: In single-deck casino blackjack, the dealer is dealt two cards from a standard deck of 52. The first card is dealt face down and the second card is dealt face up.

1. a. How many dealer hands are possible if it matters which card is face down and which is face up?
2. b. How many dealer hands are possible if it doesn’t matter which card is face down and which is face up?
3. c. Of the 52 cards in the deck, four are aces and 16 others (kings, queens, jacks, and tens) are worth 10 points each. The dealer has a blackjack if one card is an ace and the other is worth 10 points; it doesn’t matter which card is face up and which card is face down. How many different blackjack hands are there?
4. d. What is the probability that a hand is a blackjack?

To determine

Find the number of dealer hands.

Answer to Problem 40E

There are 2,652 possible dealer hands.

Explanation of Solution

Calculation:

It is given that a player dealt two cards from a deck of 52 cards.  The first card is dealt face down and second card is dealt face up. It matters which card is dealt face down and dealt face up.

Here, order matters. Hence, permutation is applied.

Permutation Rule:

The number of arrangement of r objects from n distinct objects is given by the formula:

${}_n{P}_r=\frac{n!}{\left(n-r\right)!}$

Substitute 52 for “n” and 2 for “r”.

$\begin{array}{c}N\left(\text{Dealer hands}\right){=}_{52}{P}_2\\ =\frac{52!}{\left(52-2\right)!}\\ =\frac{52!}{50!}\\ =\frac{52\times 51\times 50!}{50!}\\ =52\times 51\\ =2,652\end{array}$

Thus, there are 2,652 dealer hands are possible.

To determine

Find the number of dealer hands if it doesn’t matter which card is dealt face down and dealt face up.

Answer to Problem 40E

There are 1,326 possible dealer hands if it doesn’t matter which card is dealt face down and dealt face up.

Explanation of Solution

Calculation:

It doesn’t matter which card is dealt face down and dealt face up.

Here, order does not matter. Hence, combination is applied.

Combination Rule:

The number of combinations of r objects from n distinct objects is ${}_n{C}_r$ and is given by the formula:

${}_n{C}_r=\frac{n!}{r!\left(n-r\right)!}$

Substitute 52 for “n” and 2 for “r” in combination rule.

$\begin{array}{c}N\left(\text{dealer hands}\right){=}_{52}{C}_2\\ =\frac{52!}{2!\left(52-2\right)!}\\ =\frac{52!}{2!\times 50!}\\ =\frac{52\times 51\times 50!}{2\times 1\times 50!}\\ =\frac{52\times 51}{2\times 1}\\ =1,326\end{array}$

Thus, there are 1,326 possible dealer hands if it doesn’t matter which card is dealt face down and dealt face up.

To determine

Find the number of different blackjacks hands.

Answer to Problem 40E

There are 64 different blackjacks hands are possible.

Explanation of Solution

Calculation:

It is given that there are four aces and 16 others are worth 10 pints each. A blackjack is possible only when one card is ace and other is worth 10 points. It doesn’t matter which card is dealt face down and dealt face up.

Here, order does not matter. Hence, combination is applied.

The Fundamental Principle of Counting:

If an action can be perform in m ways and another action can be perform in n ways then both the action can be done together in mn different ways.

Number of ways an ace can be selected:

Substitute 4 for “n” and 1 for “r” in combination rule.

$\begin{array}{c}N\left(\text{ways an aces can be selected}\right){=}_4{C}_1\\ =\frac{4!}{1!\left(4-1\right)!}\\ =\frac{4!}{1!\times 3!}\\ =\frac{4\times 3!}{1\times 3!}\\ =4\end{array}$

Thus, there are 4 different ways an ace can be selected.

Number of ways a card worth 10 points can be selected:

Substitute 16 for “n” and 1 for “r” in combination rule.

$\begin{array}{c}N\left(\text{ways card worth 10 points can be selected}\right){=}_{16}{C}_1\\ =\frac{16!}{1!\left(16-1\right)!}\\ =\frac{16!}{1!\times 15!}\\ =\frac{16\times 15!}{1\times 15!}\\ =16\end{array}$

Thus, there are 16 different ways a card worth 10 points can be selected.

Substitute these values in the fundamental principle of counting.

Therefore,

$\begin{array}{c}N\left(\text{Blackjack}\right)=\left[\begin{array}{l}N\left(\text{ways an ace can be selcted}\right)\times \\ N\left(\text{ways a card worth 10 points can be selcted}\right)\end{array}\right]\\ {=}_4{C}_1{\times }_{16}{C}_1\\ =4\times 16\\ =64\end{array}$

Thus, there are 64 different blackjacks hands are possible.

To determine

Find the probability that hand is a blackjack.

Answer to Problem 40E

The probability that hand is a blackjack is 0.0483.

Explanation of Solution

Calculation:

The probability of an event can be obtained as shown below:

$P\left(A\right)=\frac{\text{number of outcomes in }A}{\text{ number of outcomes in the sample space}}$

Event A denotes that hand is a blackjack.

From part (b), it is clear that there are 1,326 possible dealer hands if it doesn’t matter which card is dealt face down and dealt face up.

From part (c), it is clear that there are 64 different blackjacks hands are possible.

Substitute 64 for “number of outcomes in A” and 1,326 for “number of outcomes in the sample space”

The required probability is obtained as follows:

$\begin{array}{c}P\left(A\right)=\frac{64}{1,326}\\ =0.04826\\ =0.0483\end{array}$

Thus, the probability that hand is a blackjack is 0.0483.