Chapter 5.3, Problem 66E

(a)

To determine

Probability for the randomly chosen return shows an adjusted gross income of $50,000 or more.

Answer to Problem 66E

Probability that the randomly chosen return shows an adjusted gross income of $50,000 or more is 0.321.

Explanation of Solution

Given information:

The distribution of the adjusted gross income (in thousands of $):

  

According to the addition rule for mutually exclusive event,

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

Note that

In the table,

Probability for the adjusted gross income of 50 − 99(in thousands of $),

  $P\left(50-99\right)=0.215$

Probability for the adjusted gross income of 100 − 499(in thousands of $),

  $P\left(100-499\right)=0.100$

Probability for the adjusted gross income of $500,000 or more,

  $P\left(\ge 500\right)=0.006$

For the probability of adjusted gross income of $50,000 or more, add up the above probabilities,

  $\begin{array}{l}P\left(\text{income}>\$50,000\right)=P\left(>50\right)\\ =P\left(50-99\right)+P(100-499)+P\left(\ge 500\right)\\ =0.215+0.100+0.006\\ =0.321\end{array}$

Thus,

For randomly chosen return, probability for adjusted gross income of $50,000 or more is 0.321.

(b)

To determine

Conditional probability for income is at least $100,000.

Answer to Problem 66E

Conditional probability for income of at least $100,000 is approx. 0.3302.

Explanation of Solution

Given information:

The distribution of the adjusted gross income (in thousands of $):

  

According to the conditional probability,

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

From Part (a):

The probability for adjusted gross income of $50,000 or more,

  $P\left(\text{income}>\$50,000\right)=0.321$

Since the return shows an income of at least $50,000.

Then

  $\begin{array}{l}P\left(\ge \$50,000\text{and}\ge \$100,000\right)=P\left(\ge \$100,000\right)\\ =P\left(100-499\right)+P\left(\ge 500\right)\\ =0.100+0.006\\ =0.106\end{array}$

Apply conditional probability:

  $\begin{array}{l}P\left(\ge \$100,000|\ge \$50,000\right)=\frac{P\left(\ge \$50,000\text{and}\ge \$1,00,000\right)}{P\left(\ge \$50,000\right)}\\ =\frac{0.106}{0.321}\\ \approx 0.3302\end{array}$

Thus,

Conditional probability that the income is at least $100,000 is approx. 0.3302.