Chapter 8.6, Problem 64E

(a)

To determine

To find: the probability that the numbers are both even.

Answer to Problem 64E

The probability of selecting both number are even from 1 to 40 is $\frac{1}{4}$ .

Explanation of Solution

Given information:

Two integers from 1 through 40 are chosen by a random number generator.

Calculation:

Even numbers from 1 through 40 is $=\left\{2,4,6,8,\mathrm{...},40\right\}$

Number of even numbers from 1 through 40 is $=20$

The probability of selecting an even number from 1 to 40 is

  $\begin{array}{c}p\left(A\right)=\frac{20}{40}\\ =\frac{1}{2}\end{array}$

So, the probability of selecting both numbers are even from 1 to 40 is

  $\begin{array}{c}p\left(A\right)\cdot p\left(A\right)=\frac{1}{2}\cdot \frac{1}{2}\\ =\frac{1}{4}\end{array}$

The probability of selecting both number are even from 1 to 40 is $\frac{1}{4}$ .

(b)

To determine

To find: the probability that one number is even and one is odd.

Answer to Problem 64E

The probability of selecting one number is even and other is odd from 1 to 40 is $\frac{1}{4}$ .

Explanation of Solution

Given information:

Two integers from 1 through 40 are chosen by a random number generator.

Calculation:

Even numbers from 1 through 40 is $=\left\{2,4,6,8,\mathrm{...},40\right\}$

Number of even numbers from 1 through 40 is $=20$

Number of odd numbers from 1 through 40 is $=20$

The probability of selecting an even number from 1 to 40 is

  $\begin{array}{c}p\left(A\right)=\frac{20}{40}\\ =\frac{1}{2}\end{array}$

The probability of selecting an odd number from 1 to 40 is

  $\begin{array}{c}p\left(B\right)=\frac{20}{40}\\ =\frac{1}{2}\end{array}$

So, the probability of selecting one number is even and other is odd from 1 to 40 is

  $\begin{array}{c}p\left(A\right)\cdot p\left(B\right)=\frac{1}{2}\cdot \frac{1}{2}\\ =\frac{1}{4}\end{array}$

The probability of selecting one number is even and other is odd from 1 to 40 is $\frac{1}{4}$ .

(c)

To determine

To find: the probability that both numbers are less than 30.

Answer to Problem 64E

The probability of selecting both numbers are less than 30 from 1 to 40 is 0.525

Explanation of Solution

Given information:

Two integers from 1 through 40 are chosen by a random number generator.

Calculation:

Number of numbers which are less than 30 from 1 through 40 is $=29$

The probability of selecting a number which are less than 30 from 1 through 40 is

  $p\left(A\right)=\frac{29}{40}$

So, the probability of selecting both numbers are less than 30 from 1 to 40 is

  $\begin{array}{c}p\left(A\right)\cdot p\left(A\right)=\frac{29}{40}\cdot \frac{29}{40}\\ =\frac{81}{1600}\\ =0.525\end{array}$

The probability of selecting both numbers are less than 30 from 1 to 40 is 0.525

(d)

To determine

To find: the probability that the same number is chosen twice.

Answer to Problem 64E

The probability of selecting same numbers twice from 1 to 40 is 0.0006.

Explanation of Solution

Given information:

Two integers from 1 through 40 are chosen by a random number generator.

Calculation:

Number of numbers from 1 through 40 is $=40$

The probability of selecting a number from 1 to 40 is

  $p\left(A\right)=\frac{1}{40}$

So, the probability of selecting same numbers twice from 1 to 40 is

  $\begin{array}{c}p\left(A\right)\cdot p\left(A\right)=\frac{1}{40}\cdot \frac{1}{40}\\ =\frac{1}{1600}\\ =0.0006\end{array}$

The probability of selecting same numbers twice from 1 to 40 is 0.0006.