Problem 34 - A First Course in Probability

Question

The second Earl of Yarborough is reported to have bet at odds of 1000 to 1 that a bridge hand of 13 cards would contain at least one card that is ten or higher. (By ten or higher we mean that a card is either a ten, a jack, a queen, a king, or an ace.) Nowadays, we call a hand that has no cards higher than 9 a Yarborough. What is the probability that a randomly selected bridge hand is a Yarborough?

Step-by-Step Solution

Answer

The probability that a randomly selected bridge hand is a Yarborough is approximately 0.0014556% or 0.000014556. This is much lower than the odds offered by the Earl of Yarborough, making the bet favorable for him.

Step 1: Find the total number of bridge hands

Using the combination formula, we can determine the total number of bridge hands by choosing 13 cards from the 52-card deck. The combination formula is \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\], where n is the total number of items, k is the number of items to be chosen, and ! denotes the factorial. In our case, n = 52, and k = 13. \[\binom{52}{13} = \frac{52!}{13!(52-13)!} = \frac{52!}{13!39!}\] Calculating this expression, we get the total number of bridge hands: \[Total \ number \ of \ bridge \ hands = \binom{52}{13} = 635,013,559,600\]

Step 2: Find the number of Yarborough hands

A Yarborough hand does not contain any card that is 10 or higher. There are 4 suits, with 9 cards in each suit below 10 (2, 3, 4, 5, 6, 7, 8, 9). Thus, there are 36 cards that can be part of a Yarborough hand. To find the number of Yarborough hands, we need to choose 13 cards from these 36 cards: \[\binom{36}{13} = \frac{36!}{13!(36-13)!} = \frac{36!}{13!23!}\] Calculating this expression, we get the number of Yarborough hands: \[Number \ of \ Yarborough \ hands = \binom{36}{13} = 9,236,080\]

Step 3: Calculate the probability of a Yarborough hand

Now that we have both the total number of bridge hands and the number of Yarborough hands, we can calculate the probability of a randomly selected bridge hand being a Yarborough hand: \[Probability = \frac{Number \ of \ Yarborough \ hands}{Total \ number \ of \ bridge \ hands} = \frac{9,236,080}{635,013,559,600}\] Simplifying the fraction, we get the probability: \[Probability = 0.000014556\] So, the probability that a randomly selected bridge hand is a Yarborough is approximately 0.0014556%. This is significantly lower than the odds offered by the Earl of Yarborough, which suggests that his bet was a profitable one for him in the long run.