Two dice are rolled. Let the random variable X denote the number that falls uppermost on the first die and let Y denote the number that falls uppermost on the second die.
(a) Find the probability distributions of X and Y.
(b) Find the probability distribution of X + Y.
x + y 2 3 4 5 6 7
P(X + Y = x + y)
x + y 8 9 10 11 12
P(X + Y = x + y)
To find the probability distributions of X and Y, we need to determine the probability of each possible outcome when rolling a single die.
Since each die has 6 sides and they are fair, the probability of rolling any number from 1 to 6 is 1/6.
(a) Probability distribution of X:
X can take values from 1 to 6. The probability of X taking each value is the same because both dice are fair.
P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) = P(X = 5) = P(X = 6) = 1/6.
(b) Probability distribution of Y:
Same as X, Y can also take values from 1 to 6, and the probabilities are equal.
P(Y = 1) = P(Y = 2) = P(Y = 3) = P(Y = 4) = P(Y = 5) = P(Y = 6) = 1/6.
To find the probability distribution of X + Y, we need to find the sum of X and Y for every possible combination.
The possible sums range from 2 (1 + 1) to 12 (6 + 6).
x + y 2 3 4 5 6 7
P(X + Y = x + y):
To determine the probability for each sum, we can count the number of ways to get that sum and divide it by the total number of possible outcomes.
For example, to get a sum of 2, the only combination is (1, 1), so:
P(X + Y = 2) = P(X = 1, Y = 1) = (1/6) * (1/6) = 1/36.
Similarly, there is only one way to get a sum of 12, which is (6, 6), so:
P(X + Y = 12) = P(X = 6, Y = 6) = (1/6) * (1/6) = 1/36.
The rest of the probabilities can be calculated in a similar manner.
x + y 8 9 10 11 12
P(X + Y = x + y):
For example, to get a sum of 8, there are five possible combinations: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2), so:
P(X + Y = 8) = P(X = 2, Y = 6) + P(X = 3, Y = 5) + P(X = 4, Y = 4) + P(X = 5, Y = 3) + P(X = 6, Y = 2) = (1/6) * (1/6) + (1/6) * (1/6) + (1/6) * (1/6) + (1/6) * (1/6) + (1/6) * (1/6) = 5/36.
Similarly, you can find the probabilities for the sums 9, 10, 11, and 12 by counting the number of combinations that yield each sum and dividing by the total number of possible outcomes.
I hope this explanation helps you understand how to find the probability distributions of X, Y, and X + Y when rolling two dice.