Chapter 6.8, Problem 17E

i.

To determine

To make a tree diagram to show all the different possible outcomes.

Answer to Problem 17E

The tree diagram with the possible outcomes can be drawn as:

  

There are total of $16$ different possible answers can be given for the two questions.

Explanation of Solution

Given:

There are $2$ questions.

Each question has $4$ choices A, B, C, D of which only $1$ choice is correct.

Calculation:

The tree diagram with the possible outcomes can be drawn as:

  

As conclude from the tree diagram there are total of $16$ different possible answers can be given for the two questions.

Hence,

There are total of $16$ different possible answers can be given for the two questions.

ii.

To determine

To find the probability.

Answer to Problem 17E

The probability of answering both question correctly is $\frac{1}{16}$ .

Explanation of Solution

Given:

There are $2$ questions.

Each question has $4$ choices A, B, C, D of which only $1$ choice is correct.

Formula used:

  $P\left(\text{event}\right)=\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}$

Calculation:

The tree diagram with the possible outcomes can be drawn as:

  

As from the tree diagram,

Total number of possible outcomes $=16$

Number of favorable outcomes of answering both questions correctly $=1$

Therefore,

  $\begin{array}{l}P\left(\text{event}\right)=\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}\\ P=\frac{1}{16}\end{array}$

Hence,

The probability of answering both question correctly is $\frac{1}{16}$ .

iii.

To determine

To answer the statement and find the probability.

Answer to Problem 17E

The number of outcomes in which answer is option A in at least one of the two question and answer given correctly is $7$ .

The Probability of answering at least one question correctly and the answer is option A is $\frac{7}{16}$ .

Explanation of Solution

Given:

There are $2$ questions.

Each question has $4$ choices A, B, C, D of which only $1$ choice is correct.

The answer of both the question is A.

Formula used:

  $P\left(\text{event}\right)=\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}$

Calculation:

The tree diagram with the possible outcomes is shown as:

  

As from the tree diagram,

The outcomes which represent option A as an answer are (A,A) (A,B) (A,C) (A,D) (B,A) (C,A) (D,A)

Therefore,

The number of outcomes in which answer is option A in atleast one of the two question and answer given correctlyis $7$ .

As from the tree diagram,

Total number of possible outcomes $=16$

Number of favorable outcomes in which at least one of the answer is option A and answer given correctly $=7$

Therefore,

  $\begin{array}{l}P\left(\text{event}\right)=\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}\\ P=\frac{7}{16}\end{array}$

Hence,

The Probability of answering at least one question correctly and the answer is option A is $\frac{7}{16}$ .