Chapter 5.7, Problem 25PSB

To determine

To find: the probability that both conclusion are true.

Answer to Problem 25PSB

The probability that both conclusions are true is 1.

Explanation of Solution

Given information:

  $ABCD$ is a parallelogram.

Formula used:

Properties of parallelogram.

  $\text{Probability of an event}\left(E\right)=\frac{\text{Number of favorable to event }\left(\text{E}\right)}{\text{Total number of possible events}}$

Calculation:

In parallelogram $ABCD$ ,

a $\overline{AB}\cong \overline{DC}\text{ is true}\left[\because \text{ From properties of Parallelogram}\right]$

b $E\text{ is the mid-point of }\overline{\text{AC }}\text{is true}\left[\because \text{ From properties of Parallelogram}\right]$

c $\angle ADC\text{ + }\angle DAB=180°\text{is true}\left[\because \text{ From properties of Parallelogram}\right]$

d $\overrightarrow{AC}\text{ bisects }\angle \text{DAB is true}\left[\because \text{ From properties of Parallelogram}\right]$

Therefore, a, b, c, d all are true.

Total number of possible ways that we can select two out of a, b, c, d is ${}^4{C}_2$

The number of possible ways that we can select two out of a, b, c, d, which are true is ${}^4{C}_2$

  $\Rightarrow {}^4{C}_2=\frac{4!}{\left(4-2\right)!\times 2!}=\frac{4\times 3\times 2\times 1}{2\times 2}=6$

Let $E$ be the event of selecting two out of a, b, c, d, which are true.

  $P\left(E\right)=\frac{\text{Number of favorable to event }\left(\text{E}\right)}{\text{Total number of possible events}}=\frac{6}{6}=1$