Chapter 10.1, Problem 24E

Solution

To calculate: The probability of the event that all six cards are from the same suit.

The probability of the event that all six cards are from the same suit is $\frac{1}{33649}$ .

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:

There is exactly one hand in this game will all spades.

The total number of hands is $C{}_6$ .

There are only six spades in the deck, so there is only one way to make a six-card hand with six spades.

Similarly, the same way goes for the other three suits, it follows:

  $\begin{array}{l}P=\frac{4}{C{}_6}\\ =\frac{4}{\frac{24!}{6!\left(24-6\right)!}}\\ =\frac{4}{\frac{24\times 23\times 22\times 21\times 20\times 19\times 18!}{6!\times 18!}}\\ =\frac{4}{\frac{24\times 23\times 22\times 21\times 20\times 19}{6\times 5\times 4\times 3\times 2\times 1}}\\ =\frac{1}{33649}\end{array}$

Hence, the probability of the event that all six cards are from the same suit is $\frac{1}{33649}$ .