Chapter 5.2, Problem 37E

(a)

To determine

Probability replacing the question marks in the table.

Answer to Problem 37E

Both the question marks should be replaced by 0.16.

Explanation of Solution

Given information:

Assignment of probabilities:

  

Let the missing probabilities be x each (both probabilities are same).

  

Criteria for valid probability model:

• For each outcome, sum of probabilities should be equal to 1.
• All probabilities should be between 0 and 1 (including both).

Now,

Sum of the probabilities shown in the table needs to be equal to 1.

  $\text{Sum}\text{of}\text{probabilities}=0.25+0.32+x+x+0.07+0.03+0.01=1$

Combine all the terms:

  $2x+0.68=1$

Subtract 0.68 from both sides:

  $2x=0.32$

Divide both sides by 2:

  $x=0.16$

Thus,

“?” in the table should be replaced by 0.16.

(b)

To determine

Probability for chosen household contains more than 2 people.

Answer to Problem 37E

Probability,

  $P\left(\text{More}\text{than}\text{2}\text{people}\right)=0.43$

Explanation of Solution

Given information:

Assignment of probabilities:

  

If the two events cannot occur at the same time, then two events are mutually exclusive or disjoint.

According to addition rule for mutually exclusive or disjoint events:

  $P\left(A\cup B\right)=P\left(A\text{or}B\right)=P\left(A\right)+P\left(B\right)$

According to complement rule,

  $P\left(A^c\right)=P\left(\text{not}A\right)=1-P\left(A\right)$

Note that

Probability of 1 person,

  $P\left(1\text{person}\right)=0.25$

Probability of 2 people,

  $P\left(2\text{people}\right)=0.32$

Since both 1 person and 2 people are two mutually exclusive events, it is impossible for a household to contain both 1 person and 2 people.

Apply the addition rule for disjoint or mutually exclusive events:

  $\begin{array}{l}P\left(\text{At}\text{most}\text{2}\text{people}\right)=P\left(1\text{person}\right)+P\left(2\text{people}\right)\\ =0.25+0.32\\ =0.57\end{array}$

Apply complement rule:

  $\begin{array}{l}P\left(\text{More}\text{than}\text{2}\text{people}\right)=1-P\left(\text{At}\text{most}\text{2}\text{people}\right)\\ =1-0.57\\ =0.43\\ =43\%\end{array}$

Thus,

Probability that the chosen household contains more than 2 people is 0.43.