Statistics: Concepts and Controversies
Statistics: Concepts and Controversies
9th Edition
ISBN: 9781464192937
Author: David S. Moore, William I. Notz
Publisher: W. H. Freeman
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Chapter 20, Problem 16E
To determine

To find: The probability model for the total number of spots on two dice.

Expert Solution & Answer
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Answer to Problem 16E

Solution: The probability model for the total number of spots on two dice is given below.

Total spots23456789101112Probability136236336436536636536436336236136

Explanation of Solution

Calculation:

There are six spots on the six faces of a dice numbered from 1 to 6. So, when two dice are rolled, the possible outcomes or the sample space is as follows:

S={(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)}

Each of the above outcomes is equally likely with probability 136.

The total number of spots will have the following sample space:

St={2,3,4,5,6,7,8,9,10,11,12}

The possible outcomes for each value of the total spots and the associated probabilities are as follows:

Total valuePossible outcomesProbability2{(1,1)}1363{(1,2),(2,1)}136+136=2364{(1,3),(3,1),(2,2)}136+136+136=3365{(1,4),(4,1),(2,3),(3,2)}136+136+136+136=4366{(1,5),(5,1),(2,4),(4,2),(3,3)}136+136+136+136+136=5367{(1,6),(6,1),(2,5),(5,2),(3,4),(4,3)}136+136+136+136+136+136=6368{(2,6),(6,2),(3,5),(5,3),(4,4)}136+136+136+136+136=5369{(3,6),(6,3),(4,5),(5,4)}136+136+136+136=43610{(4,6),(6,4),(5,5)}136+136+136=33611{(5,6),(6,5)}136+136=23612{(6,6)}136

Since each outcome is independent, the individual probabilities are summed up.

The above probability model explains the total number of spots that can appear on the up-faces of two dice and the probability associated with them.

To find: The expected value of the total number of spots appearing on two dice.

Solution: The expected value of the total number of spots is 7.

Explanation:

Calculation:

The expected value of any discrete random variable X is computed by using the following formula:

E(X)=Xx×p(X=x)

where p(X=x) denotes the probability that the random variable takes value x.

So, by using the probability model obtained above and the formula of expectation, the expected value of the total number of spots is computed as follows:

E(Total)=[(2×136)+(3×236)+(4×336)+(5×436)+(6×536)+(7×636)+(8×536)+(9×436)+(10×336)+(11×236)+(12×136)]=[136×(2+6+12+20+30+42+40+36+30+22+12)]=136×252=7

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