Final answer:
To construct the probability distribution when two seven-faced dice are rolled, list all possible sums, count the ways each sum can occur, calculate their probabilities, and then find the mean by summing the product of each sum and its probability.
Step-by-step explanation:
To construct a probability distribution for the sum shown on the faces when two dice, each with 7 faces, are rolled, we will consider all possible outcomes. The smallest sum is 2 (1+1) and the largest sum is 14 (7+7).
- Sum of 2: 1 way (1+1)
- Sum of 3: 2 ways (1+2, 2+1)
- Sum of 4: 3 ways (1+3, 2+2, 3+1), and so on up to:
- Sum of 14: 1 way (7+7)
Each die has 7 faces, so there are a total of 7*7 = 49 outcomes. To find the probability of each sum, divide the number of ways to get the sum by 49.
Probability Distribution for the Sum
The mean (μ) is calculated by multiplying each sum by its probability and then adding all these products together. The formula for the mean is:
μ = Σ (sum × probability)
After calculating the mean using the formula, round the final answer to one decimal place, but do not round during intermediate steps to ensure the accuracy of the calculation.
To find the mean of the distribution, you can use a table to list each sum and its corresponding probability, and then apply the formula for the mean.