a team of 3 is to be chosen from 4 boys and 5 girls. if X is the random variable of the number of girls in the team find the Expected variable X. E(X)
To find the expected value or expected variable (E(X)) for the random variable X, which represents the number of girls in the team, we can use the probability distribution of X.
In this case, we have 4 boys and 5 girls to choose from, and we are selecting a team of 3 members. To determine the probability distribution, we'll look at the different possibilities for the number of girls in the team: 0 girls, 1 girl, 2 girls, and 3 girls.
1. Probability of 0 girls in the team:
We need to choose all 3 members from the 4 boys, which can be done in C(4,3) ways (combinations). So, the probability is P(X=0) = C(4,3)/C(9,3).
2. Probability of 1 girl in the team:
We need to choose 1 girl from the 5 available girls and 2 boys from the 4 available boys. This can be done in C(5,1) * C(4,2) ways. So, the probability is P(X=1) = (C(5,1) * C(4,2))/C(9,3).
3. Probability of 2 girls in the team:
We need to choose 2 girls from the 5 available girls and 1 boy from the 4 available boys. This can be done in C(5,2) * C(4,1) ways. So, the probability is P(X=2) = (C(5,2) * C(4,1))/C(9,3).
4. Probability of 3 girls in the team:
We need to choose all 3 members from the 5 available girls, which can be done in C(5,3) ways. So, the probability is P(X=3) = C(5,3)/C(9,3).
Now, to calculate the expected value (E(X)), we multiply each possible outcome (number of girls) by its respective probability, and then sum them up:
E(X) = (0 * P(X=0)) + (1 * P(X=1)) + (2 * P(X=2)) + (3 * P(X=3))
Substituting the probabilities we calculated, we obtain:
E(X) = (0 * C(4,3)/C(9,3)) + (1 * (C(5,1) * C(4,2))/C(9,3)) + (2 * (C(5,2) * C(4,1))/C(9,3)) + (3 * C(5,3)/C(9,3))
Simplifying this expression will give you the answer for the expected value of X.