Sir John Suckling invented the card game cribbage in the seventeenth century. This card game is played with a standard deck. An advantageous hand in cribbage contains many cards that total to 15 in various ways. Therefore, a hand that contains fives is desirable. What is the probability of each of the following.  


  1. Being dealt 4 fives in a cribbage hand of 6 cards.
  2. Being dealt at least 1 five in a cribbage hand of 6 cards.

The order of the cards in the hand is not important; use combinations.

If a 6-card hand contains the 4 fives from the deck, the other 2 cards in the hand are selected from the remaining 48 cards. This can be done in 48 C 2 ways. There is only 1 way to choose 4 fives because 4 C 4 = 1. Therefore, the number of cribbage hands containing 4 fives is 48 C 2 × 1 = 48 C 2 .

A 6-card hand can be dealt from 52 cards in 52 C 6 ways. Therefore, the total number of possible cribbage hands is 52 C 6 .

The probability of being dealt a hand of 6 cards containing 4 fives is about 0.000 055 4 or 0.00554%.

  1. To calculate the probability of being dealt at least 1 five in a hand of 6 cards, consider the complement, a hand with no fives.

For a hand with no fives, choose the hand from the 48 cards that are not fives. A 6-card hand with no fives can be chosen in 48 C 6 ways.

As in question (a), a 6-card hand can be dealt from 52 cards in 52 C 6 ways. Therefore, the total number of possible cribbage hands is 52 C 6 .

The probability of being dealt at least 1 five in a cribbage hand is approximately 0.379 23, or about 38%.

 

Watch the Determining Probability video to see more examples of using combinations to solve probability problems.