The Basic Practice of Statistics
The Basic Practice of Statistics
8th Edition
ISBN: 9781319057916
Author: Moore
Publisher: MAC HIGHER
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Chapter 13, Problem 13.48E

(a)

To determine

To find: The probability of rolling doubles on a single toss of the dice.

(a)

Expert Solution
Check Mark

Answer to Problem 13.48E

The probability of rolling doubles on a single toss of the dice is 0.167.

Explanation of Solution

Given info:

A rolling pair of balanced dice in a board game is given and the rolls are independent of each other.

Calculation:

Two dice are rolled the sample space S is given 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)}

The probability of rolling doubles on a single toss of the dice is obtained below:

The possible outcomes to get doublets are {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}

The number of double outcomes in a pair of dice is 6 and the total number of outcomes is 36.

The required probability is as follows:

P(Rollingdoublesin asingledice)=Number of favorable casesTotal number of cases=636=16=0.167

Thus, the probability of rolling doubles on a single toss of the dice is 0.167.

(b)

To determine

To find: The probability of getting no doubles on the first toss, but doubles on the second toss.

(b)

Expert Solution
Check Mark

Answer to Problem 13.48E

The probability of getting no doubles on the first toss, but doubles on the second toss is0.139.

Explanation of Solution

Calculation:

From part (a), the probability of rolling doubles on a single toss of the dice is 0.167.

Hence, the probability of not rolling doubles on a single toss of the dice is 0.833(=10.167) 0.167.

Also, each toss is independent of the other.

The required probability is,

P(Nodoubles on the first toss and roll doubles on the second toss)=[P(Nodoubles on the first toss)P(Roll doubles on the second toss)]=0.833×0.167=0.139

Thus, the probability of getting no doubles on the first toss, but doubles on the second toss is0.139.

(c)

To determine

To find: The probability of getting no doubles in first two tosses and getting doubles in the third toss.

(c)

Expert Solution
Check Mark

Answer to Problem 13.48E

The probability of getting no doubles in first two tosses and getting doubles in the third toss is 0.1159.

Explanation of Solution

Calculation:

The required probability is,

P(Nodoubles on the first and second toss, and roll doubles on the third toss)=[P(Nodoubles on the first toss)P(Nodoubles on the second toss)P(Roll doubles on the third toss)]=0.833×0.833×0.167=0.1159

Thus, the probability of getting no doubles in first two tosses and getting doubles in the third toss is 0.1159.

(d)

To determine

To find: The probability that first doubles occurs on the fourth toss and on the fifth toss and also give the general result that the first doubles occurs on the kth toss.

(d)

Expert Solution
Check Mark

Answer to Problem 13.48E

The probability that first doubles occurs on the fourth toss is 0.0965.

The probability that first doubles occurs on the fourth toss is 0.0804.

The probability that first doubles occurs on the kth toss is (0.833)k1(0.167) .

Explanation of Solution

Given info:

Calculation:

The probability that first doubles occurs on the fourth toss is obtained below:

P(Doubles on the fourth toss)=[P(Nodoubles on the first toss)P(Nodoubles on the second toss)P(Nodoubles on the third toss)P(Doubles on the fourth toss)]=0.833×0.833×0.833×0.167=0.0965

Thus, the probability that first doubles occurs on the fourth toss is 0.0965.

The probability that first doubles occurs on the fifth toss is obtained below:

P(Doubles on the fifth toss)=[P(Nodoubles on the first toss)P(Nodoubles on the second toss)P(Nodoubles on the third toss)P(Nodoubles on the fourth toss)P(Doubles on the fifth toss)]=0.833×0.833×0.833×0.833×0.167=0.0804

Thus, the probability that first doubles occurs on the fourth toss is 0.0804.

The probability that first doubles occurs on the kth toss is obtained below:

P(Doubles on the kth toss)=[P(Nodoubles on the first toss)P(Nodoubles on the second toss)P(Nodoubles on the third toss)P(Nodoubles on the fourth toss)...P(Doubles on the kth toss)]=0.833×0.833×0.833×0.833(k1)times×0.167=(0.833)k1(0.167)

(e)

To determine

To find: The probability to get to go again within three turns.

(e)

Expert Solution
Check Mark

Answer to Problem 13.48E

The probability to get to go again within three turns is 0.4219.

Explanation of Solution

Calculation:

The probability that you get to go again within three turns is obtained below:

The required probability is as follows:

P(Go again within three turns)=0.167+0.139+0.1159

Thus, the probability to get to go again within three turns is 0.4219.

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