The numbers 1,2,…,17 are divided into 5 disjoint sets. One set has 5 elements, one set has 4 elements, two sets have 3 elements and the last set contains the 2 remaining elements. Two players each choose a number from 1 to 17 at random. The probability they choose numbers from the same set can be expressed as a/b, where a and b are coprime positive integers. What is the value of a+b?
Hmmm. For each set of size n, there's a n/17 chance that each person picked a number from that set.
(5/17)^2 + (4/17)^2 + 2(3/17)^2 + (2/17)^2 = 63/289
352
Which set or sets 1/9 belongs to
To solve this problem, let's consider the possible cases for the players choosing numbers from the same set.
Case 1: Both players choose from the set of 5 elements
In this case, the probability of the first player choosing any number from the set is 5/17. After the first player chooses a number, the second player has 4/16 = 1/4 chance of choosing a number from the same set. Therefore, the probability for this case is (5/17) * (1/4) = 5/68.
Case 2: Both players choose from one of the sets with 3 elements
There are two sets with 3 elements, so there are two ways this case can occur. For each of these sets, the first player has 3/17 probability of choosing a number from the set. After the first player chooses a number, the second player has 2/16 = 1/8 chance of choosing a number from the same set. Hence, the probability for this case is 2 * (3/17) * (1/8) = 6/136 = 3/68.
Case 3: Both players choose from the set of 4 elements
The first player has 4/17 probability of choosing a number from the set. After the first player chooses a number, the second player has 3/16 chance of choosing a number from the same set. Thus, the probability for this case is (4/17) * (3/16) = 12/272 = 3/68.
Case 4: Both players choose from the set of 2 elements
The probability of the first player choosing any number from the set is 2/17. After the first player chooses a number, the second player has 1/16 chance of choosing a number from the same set. Therefore, the probability for this case is (2/17) * (1/16) = 2/272 = 1/136.
The total probability is the sum of the probabilities for each case:
(5/68) + (3/68) + (3/68) + (1/136) = 12/68 = 3/17.
Therefore, the probability of the two players choosing numbers from the same set is 3/17. Hence, in the fraction a/b, a = 3 and b = 17. The sum of a + b is 3 + 17 = 20.
So, the value of a + b is 20.