Chapter 12.8, Problem 45PPE

a.

To determine

To find how many two digit numbers can be formed.

Answer to Problem 45PPE

  $6$

Explanation of Solution

Given:

A two digit number is formed by selecting from the digits $1,2,3,4,5$ without replacement.

Calculation:

Consider a two digit number is formed by randomly selecting the digits from 1, 2, 3, and 5 without replacement.

Using the definition of combination.

The selection of $n$ objects taking $r$ at time, where, $r\le n$ is denoted by $C(n,r)$ and defined by:

  $C(n,r)=\frac{n!}{r!(n-r)!}$ ............ (1)

Put the value $n=4,r=2$ , in the equation (1):

  $\begin{array}{l}C(4,2)=\frac{4!}{2!(4-2)!}\\ =\frac{4\times 3}{2}=6\end{array}$

Therefore, $6$ two-digit number can be formed.

b.

To determine

To find the probability that a two-digit number contains a $2$ or a $5$ .

Answer to Problem 45PPE

  $\frac{1}{6}$

Explanation of Solution

Given:

A two digit number is formed by selecting from the digits $1,2,3,4,5$ without replacement.

Calculation:

The, two-digit number contains $2$ or $5$ are not mutually exclusive event, since one two digit number is common for both event, which is $25$ .

The two-digit number contains both $2$ and $5$ number.

Therefore the event is not mutually exclusive events.

If the event A, and B are not mutually exclusive event that is overlapping event (at least one outcomes is common to both event is called overlapping event).

Then the probability is defined as:

  $P(A\text{ or}B)=P(A)+P(B)-P(A\text{ and }B)$ ............ (2)

There are $6$ two-digit number can formed, so, the probability that a two-digit number contains a $2$ or a $5$ is calculated as follows:

  $\begin{array}{l}P(2\text{ or}5)=P(2)+P(5)-P(2\text{ and 5})\\ =\frac{1}{6}+\frac{1}{6}-\frac{1}{6}\\ =\frac{1}{6}\end{array}$ ............ (3)

To the probability that a two-digit number contains a 2 or a 5 is $\frac{1}{6}$

c.

To determine

To find the probability that two-digit number is prime.

Answer to Problem 45PPE

  $\frac{1}{3}$

Explanation of Solution

Given:

A two digit number is formed by selecting from the digits $1,2,3,4,5$ without replacement.

Calculation:

Here, there are $6$ two digits number can be formed, which are $12,13,15,23,25,$ and $35.$

There are total $6$ possible outcomes.

Out of $6$ two-digit number there are two prime numbers, which is $13$ , and $23$ .

Then, the number of favorable outcomes is $2$ .

Therefore, the probability of a two digits number is prime is calculated as follows:

  $P(\text{two digit prime)}=\frac{2}{6}=\frac{1}{3}$

Thus, the probability that two-digit number is prime is $\frac{1}{3}$ .