Chapter 13.4, Problem 12PPS

To determine

To calculate:

The expected value of given dart game.

Answer to Problem 12PPS

The expected value of given dart game is $41.4$ .

Explanation of Solution

Given information:

  

In this,

Blue color has value $=25$

White color has value $=50$

Red color has value $=100$

Calculation:

A dartboard consists of $5$ regions of three different colors. The innermost circle is a red $100$ point circle, there are $2$ white wedge shaped regions that are worth $50$ points, and there are $2$ blue wedge shaped regions that are worth $25$ points.

The center red circle has a radius of $1.5$ units, and the entire circular dartboard has a radius of $6$ units.

The area of a circle is equal to $\pi r^2$ , where $r$ is the radius of the circle. Use this equation to find the probability of the red region of the circle being hit by a dart.

  $P\left(red\right)=\frac{A_r}{A_d}$

where,

  $A_r=$ Area of red region

  $A_d=$ Area of dartboard

  $\begin{array}{l}P\left(red\right)=\frac{A_r}{A_d}\\ =\frac{\pi {\left(1.5\right)}^2}{\pi {\left(6\right)}^2}\\ =\frac{2.25}{3.6}\\ =0.0625\end{array}$

Now, calculate the probability of the white region of the circle being hit by a dart.

  $\begin{array}{l}P\left(white\right)=\frac{\frac{A_d-A_r}{2}}{A_d}\\ =\frac{\frac{\pi {\left(6\right)}^2-\pi {\left(1.5\right)}^2}{2}}{\pi {\left(6\right)}^2}\\ =\frac{33.75}{2}\\ =0.46875\end{array}$

Now calculate the probability of the blue region of the circle being hit by a dart.

  $\begin{array}{l}P\left(blue\right)=\frac{\frac{A_d-A_r}{2}}{A_d}\\ =\frac{\frac{\pi {\left(6\right)}^2-\pi {\left(1.5\right)}^2}{2}}{\pi {\left(6\right)}^2}\\ =\frac{33.75}{2}\\ =0.46875\end{array}$

The total probability the red region of the circle being hit by a dart was found to be $0.0625$ , the probability that the white region of the circle being hit by a dart was found to be $0.46875$ , and probability the blue shaded of the circle being hit by a dart was found to be $0.46875$ .

Now calculate the expected value of the dartboard by adding the products of the probabilities and corresponding point value.

  $\begin{array}{l}Expectedvalue=p\left(red\right)\left(100\right)+p\left(white\right)\left(50\right)+p\left(blue\right)\left(25\right)\\ =\left(0.0625\right)\left(100\right)+\left(0.46875\right)\left(50\right)+\left(0.46875\right)\left(50\right)\\ =6.25+23.44+11.72\\ =41.4\end{array}$

To simulate the dartboard, generate random integers that correspond to each of the probabilities found for each region of the dartboard. Run $50$ trials of darts and record the frequency of observed outcomes.

The frequency of outcomes will vary between sets of $50$ trials. One such set of trials can be recorded in frequency table as follows:

Outcome	Frequency
Red	$6$
Blue	$22$
White	$22$
Total	$50$

Calculate the average value of the dartboard by adding the products of the simulated probabilities and the corresponding point values.

  $\begin{array}{l}Averagevalue=p\left(red\right)\left(100\right)+p\left(white\right)\left(50\right)+p\left(blue\right)\left(25\right)\\ =\left(\frac{6}{50}\right)\left(100\right)+\left(\frac{22}{50}\right)\left(50\right)+\left(\frac{22}{50}\right)\left(50\right)\\ =12+22+11\\ =\left|45\right|\end{array}$