From a group of 4 women and 8 men, a delegation of 3 is selected. Find the expected number of women in the delegation.
Can someone please help me find this answer?
Certainly! To find the expected number of women in the delegation, we first need to calculate the probability of selecting each possible outcome.
In this case, we have a total of 12 people, 4 women and 8 men. We want to select a delegation of 3 people.
To calculate the probability of selecting a specific number of women in the delegation, we need to consider the different ways we can choose women and men from the group.
Let's start with selecting all women in the delegation. Since there are 4 women in total, we can select 3 women from this group in only one way.
Next, let's consider selecting 2 women and 1 man in the delegation. We can choose 2 women from the 4 available in C(4,2) = 6 ways. Similarly, we can choose 1 man from the 8 available in C(8,1) = 8 ways. Thus, the total number of ways to select 2 women and 1 man is 6 * 8 = 48.
Similarly, we can calculate the number of ways to select 1 woman and 2 men in the delegation. We can choose 1 woman from the 4 available in C(4,1) = 4 ways. We can choose 2 men from the 8 available in C(8,2) = 28 ways. So, the total number of ways to select 1 woman and 2 men is 4 * 28 = 112.
Lastly, we can calculate the number of ways to select all men in the delegation. Since there are 8 men in total, we can select 3 men from this group in C(8,3) = 56 ways.
Now, we can calculate the probability of each possible outcome by dividing the number of ways to select that outcome by the total number of ways to select a delegation of 3 from the group of 12 (C(12,3) = 220 ways).
The probability of selecting all women is 1/220, the probability of selecting 2 women and 1 man is 48/220, the probability of selecting 1 woman and 2 men is 112/220, and the probability of selecting all men is 56/220.
Finally, to find the expected number of women in the delegation, we multiply each possible outcome by its corresponding probability and sum them up.
Expected number of women = (1/220 * 3) + (48/220 * 2) + (112/220 * 1) + (56/220 * 0)
= 3/220 + 96/220 + 112/220 + 0
= 211/220
≈ 0.959
So, the expected number of women in the delegation is approximately 0.959.