Chapter 8.6, Problem 8E

a.

To determine

Check whether the probability formula for the probability of the union of two events with the correct probability.

Answer to Problem 8E

The correct answer is (iii).

Explanation of Solution

Given information:

Probability of the union of two events

Formula Used:

The formula of union of two events is $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$ .

Calculation:

Write the formula of the probability of the union of two events.

Let A and B are the events in sample space.

The probability of the union of two events or probability of A or Bis :

  $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$

Hence, the correct answer is (iii).

b.

To determine

Check whether the probability formula for the probability of mutually exclusive events with the correct probability.

Answer to Problem 8E

The correct answer is (i).

Explanation of Solution

Given information:

Probability of mutually exclusive events

Formula Used:

The formula of mutually exclusive events is $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)$ .

Calculation:

Write the formula of the probability of mutually exclusive events.

Let A and B are mutually exclusive, then the formula is:

  $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)$

Hence, the correct answer is (i).

c.

To determine

Check whether the probability formula for the probability of independent events with the correct probability.

Answer to Problem 8E

The correct answer is (iv).

Explanation of Solution

Given information:

Probability of independent events

Formula Used:

The formula of independent events is $P\left(A\text{and}B\right)=P\left(A\right)\times P\left(B\right)$ .

Calculation:

Write the formula of the probability of independent events.

Let A and B are independent events, then the formula is:

  $P\left(A\text{and}B\right)=P\left(A\right)\times P\left(B\right)$

Hence, the correct answer is (iv).

d.

To determine

Check whether the probability formula for the probability of a complement with the correct probability.

Answer to Problem 8E

The correct answer is (ii).

Explanation of Solution

Given information:

Probability of a complement

Formula Used:

The formula of a complement is $P\left(A^{\prime }\right)=1-P\left(A\right)$ .

Calculation:

Write the formula of the probability of a complement.

Let A is an event, then the notation of complement of Ais $A^{\prime }$ . The formula is:

  $P\left(A\text{or}{A}^{\prime }\right)=1$

When A and $A^{\prime }$ are mutually exclusive, then the formula is:

  $P\left(A\right)+P\left(A^{\prime }\right)=1$

Thus, the probability of complement of A is:

  $P\left(A^{\prime }\right)=1-P\left(A\right)$

Hence, the correct answer is (ii).