Chapter 7.2, Problem 44PPS

a.

To determine

To find: the meaning of the probability ${\left(\frac{1}{2}\right)}^4$ .

Answer to Problem 44PPS

  ${\left(\frac{1}{2}\right)}^4$ represents the probability of getting an even number every time in 4 times rolling of a die.

Explanation of Solution

Given information :

Probability of getting an even number in a single throw of a die is $\frac{1}{2}$ .

Probability of getting an even number twice in a double throw of a die is $\frac{1}{2}\cdot \frac{1}{2}={\left(\frac{1}{2}\right)}^2$ .

Calculation:

Probability of getting an even number in a single throw of a die is $\frac{1}{2}$ .

Probability of getting an even number twice in a double throw of a die is $\frac{1}{2}\cdot \frac{1}{2}={\left(\frac{1}{2}\right)}^2$ .

So, if die is thrown 4 times and every time an even number is obtained, then the probability will be

  $\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}={\left(\frac{1}{2}\right)}^4$

So, ${\left(\frac{1}{2}\right)}^4$ represents the probability of getting an even number every time in 4 times rolling of a die.

b.

To determine

To write: an expression for the probability of getting an even number every time in $d$ times rolling of a die.

Answer to Problem 44PPS

  $2^{-m}$ represents the probability of getting an even number every time in $d$ times rolling of a die.

Explanation of Solution

Given information :

Probability of getting an even number in a single throw of a die is $\frac{1}{2}$ .

Probability of getting an even number twice in a double throw of a die is $\frac{1}{2}\cdot \frac{1}{2}={\left(\frac{1}{2}\right)}^2$ .

Calculation:

Probability of getting an even number in a single throw of a die is $\frac{1}{2}$ .

Probability of getting an even number twice in a double throw of a die is $\frac{1}{2}\cdot \frac{1}{2}={\left(\frac{1}{2}\right)}^2$ .

So, if die is thrown m times and every time an even number is obtained, then the probability will be

  $\begin{array}{c}\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \mathrm{...}\cdot \frac{1}{2}m\text{times}={\left(\frac{1}{2}\right)}^m\\ =2^{-m}\end{array}$

So, $2^{-m}$ represents the probability of getting an even number every time in $d$ times rolling of a die.