Chapter 11.3, Problem 5A

Solution

To calculate: The cumulative probabilities for a given range of sum on two fair dice when rolled.

The cumulative probabilities for the given range of sum on two fair dice when rolled are shown below

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

Given:

The table with frequencies and theoretical probabilities of the sum of numbers when two dice are rolled is shown below

Sum	Frequency	Probability
2	1	$\frac{1}{36}=0.0278$
3	2	$\frac{2}{36}=0.0556$
4	3	$\frac{3}{36}=0.0833$
5	4	$\frac{4}{36}=0.1111$
6	5	$\frac{5}{36}=0.1389$
7	6	$\frac{6}{36}=0.1667$
8	5	$\frac{5}{36}=0.1389$
9	4	$\frac{4}{36}=0.1111$
10	3	$\frac{3}{36}=0.0833$
11	2	$\frac{2}{36}=0.0556$
12	1	$\frac{1}{36}=0.0278$

Formula used:

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

Calculation:

With reference to the probability distribution, the probability that the sum is 2 to 4 is calculated as shown below

  $\begin{array}{c}P(2\text{to}4)=P(2)+P(3)+P(4)\\ =\frac{1}{36}+\frac{2}{36}+\frac{3}{36}\\ =\frac{6}{36}\end{array}$

The probability that sum is from 2 to 6 is calculated as shown below

  $\begin{array}{c}P(2\text{to}6)=P(2\text{to}4)+P(5)+P(6)\\ =\frac{6}{36}+\frac{4}{36}+\frac{5}{36}\\ =\frac{15}{36}\end{array}$

The probability that sum is from 2 to 8 is calculated as shown below

  $\begin{array}{c}P(2\text{to}8)=P(2\text{to}6)+P(7)+P(8)\\ =\frac{15}{36}+\frac{6}{36}+\frac{5}{36}\\ =\frac{26}{36}\end{array}$

The probability that sum is from 2 to 10 is calculated as shown below

  $\begin{array}{c}P(2\text{to}10)=P(2\text{to}8)+P(9)+P(10)\\ =\frac{26}{36}+\frac{4}{36}+\frac{3}{36}\\ =\frac{33}{36}\end{array}$

The probability that sum is from 2 to 12 is calculated as shown below

  $\begin{array}{c}P(2\text{to}12)=P(2\text{to}10)+P(11)+P(12)\\ =\frac{33}{36}+\frac{2}{36}+\frac{1}{36}\\ =\frac{36}{36}\end{array}$

Thus the cumulative probability distribution for given ranges of sum, when two dice are rolled is as follow:

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