From a group of 5 women and 7
men, a delegation of 3 is selected. Find the expected number of women in the delegation.
What is the expected number of women?
(Round to two decimal places as needed.)
5 women / 12 people total
= 0.4166666... probability of a woman being chosen * 3 selected = 1.25
(I think)
To find the expected number of women in the delegation, you need to calculate the probabilities of each possible outcome and then find the weighted average of these probabilities.
Step 1: Calculate the probability of each possible outcome:
There are a total of 12 people (5 women and 7 men) in the group. We want to select a delegation of 3 people.
To calculate the probability of selecting a certain number of women, we can use combinations:
- Probability of selecting 0 women: This can only happen if we select all 3 men. The number of ways to select 3 men from 7 is C(7, 3) = 35.
- Probability of selecting 1 woman: This can happen in three ways: selecting 1 woman and 2 men, selecting 2 women and 1 man, or selecting all 3 women. The number of ways to select 1 woman from 5 and 2 men from 7 is C(5, 1) * C(7, 2) = 105. Similarly, the number of ways to select 2 women from 5 and 1 man from 7 is C(5, 2) * C(7, 1) = 70. The number of ways to select all 3 women from 5 is C(5, 3) = 10.
- Probability of selecting 2 women: This can happen if we select 2 women and 1 man. The number of ways to select 2 women from 5 and 1 man from 7 is C(5, 2) * C(7, 1) = 70.
- Probability of selecting 3 women: This can only happen if we select all 3 women. The number of ways to select 3 women from 5 is C(5, 3) = 10.
Step 2: Calculate the probability of each outcome:
To calculate the probability of each outcome, divide the number of ways each outcome can occur by the total number of possible outcomes (which is the number of ways to select 3 people from 12, or C(12, 3) = 220).
Probability of selecting 0 women = 35/220 = 0.1591
Probability of selecting 1 woman = (105 + 70 + 10)/220 = 0.7045
Probability of selecting 2 women = 70/220 = 0.3182
Probability of selecting 3 women = 10/220 = 0.0455
Step 3: Calculate the expected number of women:
To calculate the expected number of women, multiply the probability of each outcome by the number of women in that outcome, and then sum up these values.
Expected number of women = (0 women * probability of selecting 0 women) + (1 woman * probability of selecting 1 woman) + (2 women * probability of selecting 2 women) + (3 women * probability of selecting 3 women)
Expected number of women = (0 * 0.1591) + (1 * 0.7045) + (2 * 0.3182) + (3 * 0.0455) ≈ 0.9545
Therefore, the expected number of women in the delegation is approximately 0.95 (rounded to two decimal places).
To find the expected number of women in the delegation, we need to calculate the probability of each possible outcome and then multiply it by the number of women in that outcome.
Step 1: Calculate the total number of ways to select a delegation of 3 from the group of 12 (5 women + 7 men). This is known as the total number of outcomes.
Total number of outcomes = (12 choose 3) = 12! / (3! * (12-3)!) = 220
Step 2: Calculate the number of outcomes where the delegation consists of no women.
Number of outcomes with no women = (7 choose 3) = 7! / (3! * (7-3)!) = 35
Step 3: Calculate the number of outcomes where the delegation consists of exactly 1 woman.
Number of outcomes with 1 woman = (5 choose 1) * (7 choose 2) = (5! / (1! * (5-1)!)) * (7! / (2! * (7-2)!)) = 5 * 21 = 105
Step 4: Calculate the number of outcomes where the delegation consists of exactly 2 women.
Number of outcomes with 2 women = (5 choose 2) * (7 choose 1) = (5! / (2! * (5-2)!)) * (7! / (1! * (7-1)!)) = 10 * 7 = 70
Step 5: Calculate the number of outcomes where the delegation consists of all 3 women.
Number of outcomes with 3 women = (5 choose 3) = 5! / (3! * (5-3)!) = 10
Step 6: Calculate the probability of each outcome by dividing the number of outcomes in that category by the total number of outcomes.
Probability of no women = Number of outcomes with no women / Total number of outcomes = 35 / 220 = 0.1591
Probability of 1 woman = Number of outcomes with 1 woman / Total number of outcomes = 105 / 220 = 0.4773
Probability of 2 women = Number of outcomes with 2 women / Total number of outcomes = 70 / 220 = 0.3182
Probability of 3 women = Number of outcomes with 3 women / Total number of outcomes = 10 / 220 = 0.0455
Step 7: Multiply the probability of each outcome by the number of women in that outcome, and sum the results to find the expected number of women.
Expected number of women = (0 * 35) + (1 * 105) + (2 * 70) + (3 * 10) = 0 + 105 + 140 + 30 = 275
Step 8: Round the expected number of women to two decimal places.
Expected number of women = 275
Therefore, the expected number of women in the delegation is 275 (rounded to two decimal places).