Question

Suppose a coin is tossed three times. Let x be a random variable whose value is 1 if the number of heads is divisible by 3, and 0 otherwise. Set up the sample space for x and the associated probabilities. Find x and σ.

Short answer

The solution has been derived as$\sigma =\frac{\sqrt{3}}{4}$

Step-by-step solution

Given Information.

It has been given that a coin is tossed three times.

Definition of Probability. 

Probability means the chances of any event to occur is called it probability.

Find the probability. 

The sample space is given as described below.

$S=\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}$

Assume x =the number of heads divisible by 3 so values of x are given by the expression mentioned below.

$x=\{0,1\}$

The probability of getting number of heads divisible by 3.

$\begin{array}{c}p(x=1)=\frac{2}{8}\\ P(x=0)=\frac{6}{8}\end{array}$

The meanis given by the expression mentioned below.

$\begin{array}{c}\mu =\sum xp\left(x\right)\\ =1\times \frac{2}{8}\\ =\frac{1}{4}\end{array}$

Write the standard deviation.

$\begin{array}{c}\mathrm{var}\left(x\right)={\sigma }^2\\ =\sum {(x-\mu )}^{2}p\left(x\right)\\ =2{(1-\frac{1}{4})}^2\left(\frac{1}{8}\right)+6{\left(\frac{1}{4}\right)}^2\left(\frac{1}{8}\right)\\ =\frac{3}{16}\end{array}$

Solve for standard deviation.

$\begin{array}{c}\sigma =\sqrt{\mathrm{var}\left(x\right)}\\ =\sqrt{\frac{3}{16}}\\ =\frac{\sqrt{3}}{4}\end{array}$

Hence, the solution has been derived as.$\sigma =\frac{\sqrt{3}}{4}$