Chapter 6.1, Problem 7E

a.

To determine

To find: The outcomes for the event A and $P\left(A\right)$ .

Answer to Problem 7E

The probability is 0.155.

Explanation of Solution

Given:

The probability distribution is:

X	1	2	3	4	5	6	7	8	9
Probability	0.301	0.176	0.125	0.097	0.079	0.067	0.058	0.051	0.046

Calculation:

Here. A be the event that depicts that the first digit is 7 or above

The outcomes for the event A can be written as:

  $\text{Outcomes = }\left\{7,8,9\right\}$

  $P\left(A\right)$ can be calculated as:

  $\begin{array}{c}P\left(A\right)=P\left(X=7\right)+P\left(X=8\right)+P\left(X=9\right)\\ =0.058+0.051+0.046\\ =0.155\end{array}$

Thus, the probability is 0.155.

b.

To determine

To find: The outcomes for the event B and $P\left(B\right)$ .

Answer to Problem 7E

The probability is 0.609.

Explanation of Solution

Calculation:

Here, B be the event that depicts that the first digit is odd

The outcomes for the event B can be written as:

  $\text{Outcomes = }\left\{1,3,5,7,9\right\}$

  $P\left(B\right)$ can be calculated as:

  $\begin{array}{c}P\left(B\right)=P\left(X=1\right)+P\left(X=3\right)+P\left(X=5\right)+P\left(X=7\right)+P\left(X=9\right)\\ =0.301+0.125+0.079+0.058+0.046\\ =0.609\end{array}$

Thus, the probability is 0.609.

c.

To determine

To find: The outcomes for the event A and B and compute $P\left(A\text{ or }B\right)$ . Also, explain the reason of $P\left(A\text{ or }B\right)$ not being equal to $P\left(A\right)+P\left(B\right)$

Answer to Problem 7E

The probability is 0.660.

Explanation of Solution

Calculation:

Using the information of above parts, the outcomes for the event A and B can be written as:

  $\text{Outcomes = }\left\{1,3,5,7,8,9\right\}$

  $P\left(A\text{ or }B\right)$ can be calculated as:

  $\begin{array}{c}P\left(A\text{ or }B\right)=P\left(X=1\right)+P\left(X=3\right)+P\left(X=5\right)+P\left(X=7\right)+P\left(X=8\right)+P\left(X=9\right)\\ =0.301+0.125+0.079+0.058+0.051+0.046\\ =0.660\end{array}$

Here, $PP\left(A\text{ or }B\right)\ne P\left(A\right)+P\left(B\right)$ because the events are not mutually exclusive.