pls help me on answeringgg my homework beloww

Question

The student council planned to have a Field trip. They're considering 3 attractions in Manila, 2 attractions in bulacan, and 4 attractions in laguna. If they are to choose 3 attractions to go to, determine the variance of the random variable representing the manila attraction.

Answer 1

We assume the random variable refers to the number of Manila attractions, and they randomly pick the 3 attractions. There are 3 + 2 + 4 = 9 total attractions of which 3 are in Manila

Let X1, X2, and X3 = 1 if a Manila site is visited in the first, second, and third selection.

Y = X1 + X2 + X3. We are seeking to find var(Y)

Then the var(Y) = E(X1 + X2 + X3)^2 - [E(X1 + X2 + X3)]^2

E(X1) = E(X2) = E(X3) = 3/9 = 1/3

E(X1+X2+X3)^2 = E(X1^2) + E(X2)^2 + E(X3)^2 + 2E(X1X2) + 2E(X1X3) + 2E(X2X3)

As each of the Xi is 0 or 1, E(Xi^2) = E(Xi) = 1/3

For i not equal j, as each of Xi and Xj is either 0 or 1, E(XiXj) = P(Xi = 1 and Xj = 1)

As we are sampling without replacement, for the probability that Xi = 1 and Xj = 1, this is the probability of selecting two Manila attractions in a sample of 2 = C(3,2)/C(9,2) = 3/36 = 1/12

Then, E(X1+X2+X3)^2 = E(X1^2) + E(X2)^2 + E(X3)^2 + 2E(X1X2) + 2E(X1X3) + 2E(X2X3) =

3 * 1/3 + 6 * 1/12 = 1 1/2

(E(X1+X2+X3))^2 = (3*1/3)^2 = 1^2 = 1

Then, var(Y) = E(X1+X2+X3)^2 - (E(X1+X2+X3))^2 = 1 1/2 - 1 = 1/2