The four players in a game of bridge are called North, South, East, and West. What is the probability that East and West have no diamonds? (I.e., what is the probability that all of the diamonds are held by North and South?)
To calculate the probability that East and West have no diamonds, we need to consider the total number of possible distributions of the diamonds among the four players.
The number of ways to distribute the 13 diamonds among the four players is given by the binomial coefficient, which can be calculated using the formula:
C(n, r) = n! / (r! * (n - r)!)
Where n is the total number of diamonds (13 in this case) and r is the number of diamonds assigned to one player or group of players (0 in this case).
To find the probability that East and West have no diamonds, we need to calculate the number of ways to distribute the diamonds among the other two players, North and South. Since all the diamonds are held by North and South, each of them can receive any number of diamonds from 0 to 13.
Let's calculate the probability step by step:
Step 1: Calculate the total number of ways to distribute the 13 diamonds among the four players:
C(13, 0) = 13! / (0! * (13 - 0)!) = 1
Step 2: Calculate the number of ways to distribute the diamonds among North and South:
We need to calculate the sum of binomial coefficients from r = 0 to r = 13:
C(13, 0) + C(13, 1) + C(13, 2) + ... + C(13, 13)
Since the binomial coefficients are symmetric, the sum simplifies to:
2^13 = 8192
Step 3: Calculate the probability that East and West have no diamonds:
The probability is calculated by dividing the number of ways where East and West have no diamonds (8192) by the total number of possible distributions (1):
Probability = 8192 / 1 = 8192
Therefore, the probability that East and West have no diamonds, and all the diamonds are held by North and South, is 1 in 8192.