Problem 3E

A section of an exam contains four True-False questions. A completed exam paper is selected at random, and the four answers are recorded.

a. List all 16 outcomes in the sample space.

b. Assuming the outcomes to be equally likely, find the probability that all the answers are the same.

c. Assuming the outcomes to be equally likely, find the probability that exactly one of the four answers is “True.”

d. Assuming the outcomes to be equally likely, find the probability that at most one of the four answers is “True.”

Solution 3E

Step1 of 3:

We have Exam it contains four True-False questions. Here A completed exam paper is selected at random, and the four answers are recorded.

We need to find,

a).List all 16 outcomes in the sample space.

b).Assuming the outcomes to be equally likely, find the probability that all the answers are the same.

c).Assuming the outcomes to be equally likely, find the probability that exactly one of the four answers is “True.”

d).Assuming the outcomes to be equally likely, find the probability that at most one of the four answers is “True.”

Step2 of 3:

a).

Here, we can select either true or false 4 times for one completed answer.

The possible number of outcomes are S = {TTTT, TTTF, TTFT, TFTT, FTTT,FFTT, FTFT, FTTF, TFFT, TFTF, TTFF, TFFF, FFFT, FFTF, FTFF, FFFF}.

b).

From Part(a) we have total outcomes = 16, in this total outcomes there are only 2 outcomes giving same answers and they are {TTTT, FFFF}.

Now,

The probability that all the answers are the same is given by

P(the probability that all the answers are the same) =

=

=

Therefore, The probability that all the answers are the same is

Step3 of 3:

c).

From Part(a) we have total outcomes = 16, in this total outcomes there are only 4 outcomes having one True in the answers and they are {TFFF, FFFT, FFTF, FTFF}.

Now,

The probability that exactly one of the four answers is “True” is given by

P(the probability that exactly one of the four answers is “True”) =

=

=

Therefore, The probability that exactly one of the four answers is “True” is

d).

From Part(a) we have total outcomes = 16, in this total outcomes there are only 5 outcomes having at most one of the four answers is “True and they are {TFFF, FFFT, FFTF, FTFF, FFFF}.

Now,

The probability that at most one of the four answers is “True.” is given by

P(at most one of the four answers is “True) =

=

Therefore, The probability that at most one of the four answers is “True.” is

Conclusion:

Therefore,

a).The possible number of outcomes are S = {TTTT, TTTF, TTFT, TFTT, FTTT,FFTT, FTFT, FTTF, TFFT, TFTF, TTFF, TFFF, FFFT, FFTF, FTFF, FFFF}.

b).The probability that all the answers are the same is

c).The probability that exactly one of the four answers is “True” is

d).The probability that at most one of the four answers is “True.” is