Final answer:
The probability distribution of the random variable x, representing the smaller number when rolling two identical six-sided dice, is determined by the outcomes where the dice show doubles. Each double has an equal probability of 1/36, leading to a symmetrical distribution for x with values from 1 to 6.
Step-by-step explanation:
The question concerns the probability distribution of a random variable x, which in this context is defined as the smaller number obtained when rolling two six-sided dice. Since the question specifies that both dice come up the same number for x to equal that value, we are interested in the outcomes where the dice show doubles (1-1, 2-2, 3-3, 4-4, 5-5, or 6-6).
There are a total of 36 possible outcomes when rolling two dice (6 options for the first die and 6 options for the second die). Each of these outcomes is equally likely. The probability of rolling a particular double is therefore 1 out of 36, or ⅔.
To find the probability distribution of x, we list the possible values of x and their associated probabilities:
- P(x=1) = 1/36
- P(x=2) = 1/36
- P(x=3) = 1/36
- P(x=4) = 1/36
- P(x=5) = 1/36
- P(x=6) = 1/36
The distribution of x will therefore have an equal probability of 1/36 for each value from 1 to 6. When summarizing the probability distribution of x, it is helpful to note the symmetrical nature of the distribution, with each outcome (doubles) having the same probability of occurrence.