Question

Tossing dice. Consider the experiment of tossing a pair of dice. Let X be the outcome (i.e., the number of dots appearing face up) on the first die and let Y be the outcome on the second die.

find the joint probability distribution p(x,y).

Answer 1

The joint probability distribution p(x, y) for tossing a pair of fair six-sided dice is 1/36 for each possible outcome pair (x, y), since each outcome of a die is independent and they are all equally likely.

Step-by-step explanation:

When considering the experiment of tossing a pair of fair six-sided dice, we are asked to find the joint probability distribution p(x, y). Assuming that the dice are fair, the result of each die is independent of the other, and each face (1 through 6) is equally likely to occur. The sample space for a single die roll is {1, 2, 3, 4, 5, 6}.

To find the joint probability distribution, we take the product of the individual probabilities of each die because the rolls are independent. Since each face of a fair die has a probability of 1/6, the probability p(x, y) for any particular combination (x, y) where x and y are the outcomes on the first and second die respectively, is calculated as (1/6) * (1/6) = 1/36. Therefore, the joint probability distribution for any pair (x, y), where x and y range from 1 to 6, is 1/36.

For example, the probability of rolling a 2 on the first die and a 5 on the second die (p(2,5)) is 1/36, the same as any other specific outcome pair (x, y).