Question

A fair four-sided die has two faces numbered 0 and two faces numbered 2. Another fair four-sided die has its faces numbered 0, 1, 4, and 5. The two dice are rolled. Let X and Y be the respective outcomes of the roll. Let W = X + Y. Determine the PMF of W.

Answer 1

The PMF of W can be determined by finding the probabilities of each possible sum of the dice rolls. The PMF of W is as follows: W=0: 1/8, W=1: 1/4, W=2: 1/4, W=4: 1/8, W=5: 1/4, W=6: 1/8, W=7: 1/8.

Step-by-step explanation:

The PMF (Probability Mass Function) of W can be determined by adding the values of X and Y and finding the probabilities for each possible sum. Since X can be 0 or 2 and Y can be 0, 1, 4, or 5, the possible sums of X and Y are 0, 1, 2, 4, 5, 6, and 7. To find the PMF of W, we need to calculate the probability of each sum.

1. If W = 0, the possible combinations are (X=0, Y=0). This can occur with a probability of (1/2) * (1/4) = 1/8.
2. If W = 1, the possible combinations are (X=0, Y=1) and (X=2, Y=0). Each combination can occur with a probability of (1/2) * (1/4) = 1/8. So, the total probability of W = 1 is 1/8 + 1/8 = 1/4.
3. If W = 2, the possible combinations are (X=2, Y=0) and (X=0, Y=2). Each combination can occur with a probability of (1/2) * (1/4) = 1/8. So, the total probability of W = 2 is 1/8 + 1/8 = 1/4.
4. If W = 4, the possible combination is (X=2, Y=2), which can occur with a probability of (1/2) * (1/4) = 1/8.
5. If W = 5, the possible combinations are (X=0, Y=5) and (X=2, Y=3). Each combination can occur with a probability of (1/2) * (1/4) = 1/8. So, the total probability of W = 5 is 1/8 + 1/8 = 1/4.
6. If W = 6, the possible combination is (X=2, Y=4), which can occur with a probability of (1/2) * (1/4) = 1/8.
7. If W = 7, the possible combination is (X=2, Y=5), which can occur with a probability of (1/2) * (1/4) = 1/8.

So, the PMF of W is as follows:
W=0: 1/8
W=1: 1/4
W=2: 1/4
W=4: 1/8
W=5: 1/4
W=6: 1/8
W=7: 1/8