Question

Chance of winning blackjack. Blackjack, a favourite gameof gamblers, is played by a dealer and at least one opponent (called a player). In one game version, 2 cards of a standard 52-card bridge deck are dealt to the player and 2 cards to the dealer.This exercise assumes that drawing an ace and a face card is called blackjack. If the dealer does not draw blackjack and the player does, the player wins. If the dealer and player draw blackjack, a “push” (i.e., a tie) occurs.

a. What is the probability that the dealer will draw a blackjack?

b. What is the probability that the player wins with a blackjack?

Short answer

1. The probability that the dealer will draw a blackjack is 0.0362.
2. The probability that the player wins with blackjack is 0.0352.

Step-by-step solution

Important formula

The combination formula is$$\begin{gathered} \left(\mathbf{N} \\ \mathbf{r} \\ \right)\mathbf{=}\frac{\mathbf{N}\mathbf{!}}{\mathbf{r}\mathbf{!}\mathbf{(}\mathbf{N}-\mathbf{r}\mathbf{)}\mathbf{!}} \end{gathered}$$

(a) Find the probability that the dealer will draw a blackjack

According to the given information, since an ace can be drawn 4 times and a face card can be drawn 12 times, then $\left(4\right)\left(12\right)=48$

So, there are 48 different ways.

Here total cards are 52, and 2 two cards are drawn. To get the result use a combination formula.

$$\begin{gathered} \begin{array}{rcl}\left(N \\ r \\ \right)& =& \frac{N!}{r! \left(N-r\right)!}\\ & =& \frac{52!}{2!\left(52-2\right)! \\ =\frac{52!}{2!50!} \\ =1326 \\ }\\ & & \end{array} \end{gathered}$$

$$\begin{gathered} \mathbf{P}\mathbf{\left(}\mathbf{DB}\mathbf{\right)}\mathbf{=}\frac{\mathbf{48}}{\mathbf{1326}} \\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0362} \end{gathered}$$

Hence, the probability that the dealer will draw a blackjack is 0.0362

(b) Find the probability that the player wins with a blackjack

To the given information, an ace can be drawn 4 times, and a 3-face card can be drawn 11 times$\left(3\right)\left(11\right)=33$

So, there are 33 different ways.

Here total cards are 50, and 2 two cards are drawn. To get the result use a combination formula.

$$\begin{gathered} \begin{array}{rcl}\left(N \\ r \\ \right)& =& \frac{N!}{r! \left(N-r\right)!}\\ & =& \frac{50!}{2!\left(50-2\right)!}\\ & =& \frac{52!}{2! 48!}\\ & =& 1225\\ & & \end{array} \end{gathered}$$

Now,

$\begin{array}{rcl}P\left(B{J}^C|PLAYER\right)& =& 1-\frac{33}{1225}\\ & =& \frac{1192}{1225}\\ & =& 0.973\\ & & \end{array}$

$$\begin{gathered} \mathbf{P}\mathbf{(}\mathbf{player}\cap {\mathbf{Dealernotdrawablackjck}}^{\mathbf{c}}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0362}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{973}\mathbf{)} \\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0352} \end{gathered}$$

Therefore, the probability that the player wins with blackjack is 0.0352.