Answer:
To find the mean and variance of the random variable X that describes the number of "Viclovers" per team, we need to make a few assumptions:
1. We'll assume that teams are formed randomly without replacement from the pool of 80 people.
2. Since 2 or more "Viclovers" cannot be in a group, each team can have at most 1 "Viclover."
Now, let's calculate the mean and variance:
1. Mean (Expected Value):
The probability of selecting a "Viclover" for a team is 5/80 (since there are 5 "Viclovers" out of 80 people). Therefore, the probability of not selecting a "Viclover" is (80-5)/80 = 75/80. The expected value of X can be calculated as follows:
E(X) = (0 * P(X=0)) + (1 * P(X=1))
= (0 * 75/80) + (1 * 5/80)
= 5/80
So, the mean number of "Viclovers" per team is 5/80.
2. Variance:
The variance of a random variable X can be calculated as follows:
Var(X) = E(X^2) - (E(X))^2
We've already calculated E(X) as 5/80. Now, we need to calculate E(X^2). Since X can only take values 0 or 1, we have:
E(X^2) = (0^2 * P(X=0)) + (1^2 * P(X=1))
= (0 * 75/80) + (1 * 5/80)
= 5/80
Now, calculate the variance:
Var(X) = E(X^2) - (E(X))^2
= (5/80) - (5/80)^2
So, the variance of X is (5/80) - (5/80)^2.
Interpretation:
The mean (expected value) of X is 5/80, which means on average, you can expect to have 5 "Viclovers" in every 80-person team.
The variance represents the spread or variability of X. In this case, the variance is relatively small due to the restriction that only one "Viclover" can be in a team.
Now, let's calculate the probability that 2 or more "Viclovers" cannot be in a group:
P(X ≥ 2) = P(X=2) + P(X=3) + ... + P(X=5)
Since each team can have at most 1 "Viclover," the probability of having 2 or more "Viclovers" in a team is zero:
P(X ≥ 2) = 0
So, the probability of having 2 or more "Viclovers" in a team is zero, given the specified constraints.