Chapter 11.3, Problem 6A

Solution

To explain: The cumulative probabilities for the last interval in the given range of sum on two fair dice when rolled is 1.

Given:

The cumulative probabilities of the sum of numbers when two dice are rolled is shown below

Sum	2 to 4	2 to 6	2 to 8	2 to 10	2 to 12
Cumulative Probability	$\frac{6}{36}$	$\frac{15}{36}$	$\frac{26}{36}$	$\frac{33}{36}$	$\frac{36}{36}$

Formula used:

The cumulative probability for x is calculated by adding all probabilities less than or equal to x.

Explanation:

The theoretical probability is the ratio of favourable outcomes to total possible outcomes. While the cumulative probabilities for a particular range are the sum of all probabilities of possible events in that particular range.

The last range in the cumulative probabilities of the sum obtained when two dice are rolled is from 2 to 12. The sample space for the sum is also 2 to 12, thus the range of the last interval 2 to 12 covers all the possible outcomes, so the probability is 1 as the favourable outcomes are the same as total possible outcomes.

Conclusion:From the definition of probability and cumulative probability, the cumulative probability of the last interval for any probability distribution is always 1.