Chapter 10.1, Problem 23E

(a)

To determine

To list: the possible sums that result from rolling two six-sided dice.

Answer to Problem 23E

The possible sums are: 2,3,4,5,6,7,8,9,10,11,12 .

Explanation of Solution

Given:

	A	B	C
1	First die	Second die	sum
2	4	6	10
3	3	5	9
4	1	8	7

Calculation:

A probability experiment is an action, or trial that has varying results. The possible result of a probability experiment is known as outcomes. The set of all possible outcomes is called sample space. The sample space for rolling two six sided dice are as follows:

(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)

(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)

Studying the outcomes, the possible sums that result from rolling two six sided dice are as follows:

2,3,4,5,6,7,8,9,10,11,12

Therefore, the answers are: 2,3,4,5,6,7,8,9,10,11,12 .

Conclusion:

Therefore, the possible sums are: 2,3,4,5,6,7,8,9,10,11,12 .

(b)

To determine

To find: the theoretical probability of rolling each sum.

Answer to Problem 23E

Sum	probabilities	sum	probabilities
2	0.028	8	0.1389
3	0.056	9	0.1111
4	0.0833	10	0.0833
5	0.1111	11	0.0556
6	0.1389	12	0.0278
7	0.1667

Explanation of Solution

Calculation:

There are 36 outcomes and outcomes for a specified event are called favorable outcomes. When all outcomes are equally likely, the theoretical probability is described as the ratio of number of favorable outcomes to total number of outcomes. Therefore, the required probability is as follows:

  $\begin{array}{l}P(\text{ sum is }2)=\frac{\text{ Number of favorable outcomes }}{\text{ Total number of outcomes }}\\ =\frac{1}{36}\\ =0.028\end{array}$

Equivalently, compute the remaining probabilities. The following tale illustrates the probabilities of remaining sums.

Sum	probabilities	sum	probabilities
2	0.028	8	0.1389
3	0.056	9	0.1111
4	0.0833	10	0.0833
5	0.1111	11	0.0556
6	0.1389	12	0.0278
7	0.1667

Table illustrates the probabilities of rolling each sum in the second column.

Conclusion:

Therefore, the table illustrates the probabilities of rolling each sum in the second column.

Sum	probabilities	sum	probabilities
2	0.028	8	0.1389
3	0.056	9	0.1111
4	0.0833	10	0.0833
5	0.1111	11	0.0556
6	0.1389	12	0.0278
7	0.1667

(c)

To determine

To compare: the experimental probabilities of rolling each sum with the theoretical probabilities.

Answer to Problem 23E

The results show that both are close but not exactly equal.

Explanation of Solution

Calculation:

The experimental probability is the ratio of number of successes to number of trials. Use random number generator to generate 50 numbers each ranging from 1 to 6 for two dices.

Calculate the sum. Then compute the experimental probability of each sum.

The following screen shot illustrates the probabilities of sums for first 24 trials.

  

Compare both experimental and theoretical probabilities.

The results show that both are close but not exactly equal.

Conclusion:

The results show that both are close but not exactly equal.