Assume that there are 18 board
members: 12 females, and 6 males including Larry.
There are 3 tasks to be assigned. Note that assigning the same
people different tasks constitutes a different assignment.
Find the probability that both males and females are given a task.
Find the probability that Larry and at least one female
are given tasks.
To find these probabilities, we need to understand the total number of possible assignments and then determine the number of favorable outcomes.
1. Probability that both males and females are given a task:
First, we need to find the total number of possible assignments. Since there are 18 board members and 3 tasks, we can use the combination formula:
Total possible assignments = C(18, 3) = 18! / (3! * (18-3)!) = 816
Now, let's find the number of favorable outcomes where both males and females are given a task.
Number of favorable outcomes = Number of ways to select 1 male from 6 males * Number of ways to select 2 females from 12 females
Number of favorable outcomes = C(6, 1) * C(12, 2) = 6 * (12!/(2! * (12-2)!)) = 6 * (12 * 11 / 2) = 396
Therefore, the probability that both males and females are given a task is:
P(both males and females are given a task) = Number of favorable outcomes / Total possible assignments = 396 / 816 ≈ 0.485.
2. Probability that Larry and at least one female are given tasks:
To find this probability, we first calculate the number of favorable outcomes, and then divide it by the total possible assignments.
Number of favorable outcomes = Number of ways to select Larry * Number of ways to select at least one female from the remaining 11 females
Number of favorable outcomes = C(1, 1) * (2^11 - 1) = 1 * (2048 - 1) = 2047
In this case, "2^11 - 1" represents the number of ways to select at least one female from 11 females (excluding the case where no female is selected).
Therefore, the probability that Larry and at least one female are given tasks is:
P(Larry and at least one female are given tasks) = Number of favorable outcomes / Total possible assignments = 2047 / 816 ≈ 2.51.
The same question came up back in 2009
http://www.jiskha.com/display.cgi?id=1236051521
For that question there were 4 tasks instead of your 3
make the necessary adjustments to drwls solution.