Chapter 11.3, Problem 18PPS

(a)

To determine

To find: The complete table.

Answer to Problem 18PPS

The complete table is shown in table 1.

Explanation of Solution

Given;

The radius of the spinner is 2.5 inches.

Calculation:

The formula for the area is given as,

  $A=\pi r^2\left(\frac{\theta }{360}\right)$

Complete the table.

Color	Probability	Sector Area	Total Area	Ratio
Red	(1/6)	3.27 in2	19.63 in2	0.166
Orange	(1/6)	3.27 in2	19.63 in2	0.166
Yellow	(1/6)	3.27 in2	19.63 in2	0.166
Green	(1/4)	4.91 in2	19.63 in2	0.25
Blue	(1/4)	4.91 in2	19.63 in2	0.25

Table 1

Thus, the complete table is shown in table 1.

(b)

To determine

To find: The relationship between the ratio of the area of the sector and the total area and the probability of the spinner landing on each color.

Answer to Problem 18PPS

The probability is equal.

Explanation of Solution

Consider the table (1) I part (a),

The probability is equal to the ratio of sector area to the total area.

Thus, the probability is equal.

(c)

To determine

To find: The prediction of the probability of a dart landing in each area of the board.

Answer to Problem 18PPS

The probability of a dart landing in each area of the board is calculated.

Explanation of Solution

Calculate the probability for the green area.

  $\begin{array}{c}{P}_{green}=\frac{\pi {\left(3\right)}^2}{\pi {\left(9\right)}^2}\\ =\frac{9}{81}\\ =\frac{1}{9}\end{array}$

Calculate the probability for the blue area.

  $\begin{array}{c}{P}_{blue}=\frac{\pi {\left(9\right)}^2-\pi {\left(6\right)}^2}{\pi {\left(9\right)}^2}\\ =\frac{81-36}{81}\\ =\frac{5}{9}\end{array}$

Calculate the probability for the yellow area.

  $\begin{array}{c}{P}_{yellow}=1-\left(\frac{1}{9}+\frac{5}{9}\right)\\ =\frac{1}{3}\end{array}$

Thus, the probability of a dart landing in each area of the board is calculated.

(d)

To determine

To find: The relative frequency table for throwing 100 darts.

Answer to Problem 18PPS

The relative frequency table for throwing 100 darts is shown in table 2.

Explanation of Solution

The table is drawn as,

Color	Frequency	Relative Frequency
Green	9	0.09
Yellow	36	0.36
Blue	55	0.55

Table 2

Thus, the relative frequency table for throwing 100 darts is shown in table 2.

(e)

To determine

To find: The graph for probability distribution.

Answer to Problem 18PPS

The graph for probability distribution is shown in figure 1.

Explanation of Solution

From Table 2, the graph is drawn as,

  

Thus, the graph for probability distribution is shown in figure 1.