Chapter 11.4, Problem 4GP

a.

To determine

To find the probabilities associated with the number of answers Jessica answered correctly.

Explanation of Solution

Given:

Jessica’s civics quiz consisted of five true or false questions.

Calculation:

The probability of $k$ successes in $n$ trials of a binomial experiment is,

  $P(k)=C(n,k)p^k{q}^{n-k}$

Here p represents the probability of success and q represents the probability of failure.

Since there are true or false questions.

  $p=0.5$

While the failure is

  $q=0.5$

Substitute in the above formula to find the probability that none of the answers are correct.

  $P(0)=C(5,0){(0.5)}^0{(0.5)}^5=0.03125$

Substitute in the above formula to find the probability that one of the answers are correct.

  $P(1)=C(5,1){(0.5)}^1{(0.5)}^4=0.15625$

Substitute in the above formula to find the probability that two of the answers are correct.

  $P(2)=C(5,2){(0.5)}^2{(0.5)}^3=0.3125$

Substitute in the above formula to find the probability that three of the answers are correct.

  $P(3)=C(5,3){(0.5)}^3{(0.5)}^2=0.3125$

Substitute in the above formula to find the probability that four of the answers are correct.

  $P(4)=C(5,4){(0.5)}^4{(0.5)}^1=0.15625$

Thus, the probability that all five answers are correct.

  $P(5)=C(5,5){(0.5)}^5{(0.5)}^0=0.03125$

b.

To determine

To find the probability that Jessica passes the quiz.

Answer to Problem 4GP

  $0.1875$

Explanation of Solution

Given:

Jessica’s civics quiz consisted of five true or false questions.

Calculation:

Jessica must answer at least $4$ questions correctly to pass the quiz.

So, the probability that Jessica passes the quiz is

  $\begin{array}{l}P(\text{at least 4})=P(4)+P(5)\\ =0.15625+0.03125\\ =0.1875\end{array}$

Thus, the probability that Jessica passes the quiz is $0.1875$ .

c.

To determine

To find the number of questions that Jessica expected to answer correctly.

Answer to Problem 4GP

  $3$

Explanation of Solution

Given:

Jessica’s civics quiz consisted of five true or false questions.

Calculation:

In general, the mean, of any binomial distribution is,

  $\mu =np$

Here n is the number of trials and p is the probability of success.

For the above situation, it is clear that $n=5,p=0.5$ . Substitute these values in the above formula to find the number of questions that Jessica expected to answer correctly.

  $\begin{array}{l}\mu =np\\ \mu =5\times 0.5\\ \mu =2.5\\ \mu \approx 3\end{array}$

Thus, the number of questions that Jessica expected to answer correctly is $3$ .