An executive committee consists of 11 members: 5 men and 6 women. 3 members are selected at random to attend a meeting in Hawaii. The names are drawn from a hat. What is the probability that all 3 selected are women?
Thanks for the help
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
6/11 * (6-1)/(11-1) * (6-2)/(11-2) = ?
To find the probability that all 3 selected members are women, we need to determine the total number of possible outcomes and the number of favorable outcomes.
There are 11 members in the executive committee, and we need to select 3 members. The total number of possible outcomes is given by the combination formula:
nCr = n! / (r! * (n - r)!)
Where n is the total number of members (11) and r is the number of members to be selected (3).
Applying this formula:
11C3 = 11! / (3! * (11 - 3)!) = 11! / (3! * 8!) = (11 * 10 * 9) / (3 * 2 * 1) = 165.
So, there are 165 possible combinations of selecting 3 members from the committee.
Since there are 6 women in the committee, the number of favorable outcomes (selecting all 3 women) is given by:
6C3 = 6! / (3! * (6 - 3)!) = 6! / (3! * 3!) = (6 * 5 * 4) / (3 * 2 * 1) = 20.
Therefore, the probability of selecting all 3 women is:
P(all 3 women) = Favorable outcomes / Total outcomes = 20 / 165 = 4 / 33.
Hence, the probability that all 3 selected members are women is 4/33.
To find the probability of selecting all 3 women from the executive committee, we need to consider the total number of possible outcomes and the specific desired outcome.
Total number of possible outcomes:
Since we are selecting 3 members from a committee of 11, there are 11C3 (11 choose 3) possible combinations of members that can be selected. This can be calculated using the formula:
11C3 = (11!)/(3!(11-3)!) = (11!)/(3!8!) = (11 × 10 × 9)/(3 × 2 × 1) = 165
Desired outcome:
We want to select all 3 women, which means choosing from the 6 women out of the total 11 committee members. So, we need to calculate 6C3 (6 choose 3), which is the number of combinations of 3 members that can be selected from the 6 women, using the formula:
6C3 = (6!)/(3!(6-3)!) = (6!)/(3!3!) = (6 × 5 × 4)/(3 × 2 × 1) = 20
Probability:
The probability of selecting 3 women from the executive committee can be found by dividing the desired outcome by the total number of possible outcomes:
Probability = (Desired Outcome) / (Total Outcomes)
= 6C3 / 11C3
= 20 / 165
≈ 0.1212
≈ 12.12%
Therefore, the probability that all 3 selected members will be women is approximately 12.12%.