Question

Darnell and Jake are playing a game they made up which includes throwing darts at this target.

The point values for each region of the target are shown. The radius of the 50-point region is 3 inches. The width of each of the other regions is 4 inches.
5 concentric circles where each circle contains values from 10, 20, 30, 40 and 50.

Darnell and Jake are currently tied in their game, and Jake has to choose one of these options:

Option 1: He can accept the tie, and the game is over.
Option 2: He can make one more throw. Jake wins if he earns at least 30 points on the throw, but he loses if he earns less than 30 points.

Complete each part of this task to determine which option Jake should choose.
Part A

Describe the process you could use to find the probability that Jake will earn at least 30 points on a throw, given that he hits the target. Note: Assume that it is equally likely that he will hit any region in the target.

Answer 1

Option 2: He can take one final throw and try to win the game.

If Jake chooses Option 2 and takes one final throw we need to determine the probability of him winning the game.

To calculate this probability we need to understand the scoring system of the game. The target has five concentric circles with point values of 10 20 30 40 and 50. The radius of the 50-point region is 3 inches and the width of each of the other regions is 4 inches.

Assuming Jake's aim is perfect and his throw hits the target we need to find the probability of his throw landing in the 50-point region which would make him the winner.

Since the 50-point region has a radius of 3 inches we can calculate the area of this region using the formula for the area of a circle:

A = π * r^2

A = π * (3 inches)^2

A ≈ 28.27 square inches

The total area of the target can be calculated by subtracting the area of the 50-point region from the area of the entire target.

Considering that the target has a width of 4 inches for each of the other regions the radius of the second circle (with a point value of 40) would be 3 + 4 = 7 inches.

Using the same formula for the area of a circle we can calculate the area of the second circle and subtract it from the total area of the target.

Repeat this process for the remaining circles and finally subtract the area of the 50-point region.

Let's calculate the probabilities for each region:

Probability of landing in the 50-point region = Area of 50-point region / Total area of the target

Probability of landing in the 40-point region = (Area of 40-point region - Area of 50-point region) / Total area of the target

Probability of landing in the 30-point region = (Area of 30-point region - Area of 40-point region) / Total area of the target

Probability of landing in the 20-point region = (Area of 20-point region - Area of 30-point region) / Total area of the target

Probability of landing in the 10-point region = (Area of 10-point region - Area of 20-point region) / Total area of the target

Once we calculate these probabilities we can determine Jake's chances of winning the game if he chooses Option 2 and takes one final throw.