Chapter 10.1, Problem 25E

Solution

To calculate: The probability of the event that a hand includes all four aces.

The probability of the event that a hand includes all four aces is $\frac{5}{3542}$ .

Given information:

A version of the card game “bid Euchre” that uses a pack of 24 cards, consisting of ace, king, queen, jack, 10, and 9 in each of the four suits (spades, hearts, diamonds, and clubs). In bid Euchre, a hand consists of 6 cards.

Calculation:

The number of events that can happen is $C{}_6$ .

There are four aces in the deck and six cards in one hand so that leaves $2$ more cards of $20$ possible. That leaves $C$ combinations.

The possibility of getting four aces is:

  $\begin{array}{l}\frac{C{}_2}{C{}_6}=\frac{\frac{20!}{2!\left(20-2\right)!}}{\frac{24!}{6!\left(24-6\right)!}}\\ =\frac{\frac{20\times 19\times 18!}{2\times 1\times 18!}}{\frac{24\times 23\times 22\times 21\times 20\times 19\times 18!}{6\times 5\times 4\times 3\times 2\times 1\times 18!}}\\ =\frac{\frac{20\times 19}{2\times 1}}{\frac{24\times 23\times 22\times 21\times 20\times 19}{6\times 5\times 4\times 3\times 2}}\\ =\frac{190}{134596}\\ =\frac{5}{3542}\end{array}$

The probability of the event that a hand includes all four aces is $\frac{5}{3542}$ .