To determine the probability that Jake will earn at least 30 points on a throw, given that he hits the target, we need to follow these steps:

1. Identify the total area of the target: Calculate the area of each concentric circle and sum them up. This will give us the total area where Jake's dart can land.

2. Identify the areas of the regions that score 30 points or more: Calculate the area of the regions corresponding to 30, 40, and 50 points.

3. Calculate the probability: The probability that Jake will earn at least 30 points is the ratio of the combined area of the 30-point, 40-point, and 50-point regions to the total area of the target.

Let's break down these steps in detail.

Step 1: Calculate the Total Area of the Target

The target consists of 5 concentric circles, with the radius of the 50-point region given as 3 inches and the width of each subsequent region as 4 inches.

• Radius of 50-point region: $r_5 = 3$ inches
• Radius of 40-point region: $r_4 = r_5 + 4 = 3 + 4 = 7$ inches
• Radius of 30-point region: $r_3 = r_4 + 4 = 7 + 4 = 11$ inches
• Radius of 20-point region: $r_2 = r_3 + 4 = 11 + 4 = 15$ inches
• Radius of 10-point region: $r_1 = r_2 + 4 = 15 + 4 = 19$ inches

The area of a circle is given by $\pi r^2$.

• Area of 10-point region (entire circle): $A_1 = \pi (19)^2 = 361\pi$
• Area of 20-point region: $A_2 = \pi (15)^2 = 225\pi$
• Area of 30-point region: $A_3 = \pi (11)^2 = 121\pi$
• Area of 40-point region: $A_4 = \pi (7)^2 = 49\pi$
• Area of 50-point region: $A_5 = \pi (3)^2 = 9\pi$

Step 2: Calculate the Areas of the Regions that Score 30 Points or More

We need to calculate the area for the regions corresponding to 30, 40, and 50 points:

• Area of 30-point region: $A_{30} = A_3 - A_4 = 121\pi - 49\pi = 72\pi$
• Area of 40-point region: $A_{40} = A_4 - A_5 = 49\pi - 9\pi = 40\pi$
• Area of 50-point region: $A_{50} = A_5 = 9\pi$

The combined area for regions scoring at least 30 points: $A_{\geq 30} = A_{30} + A_{40} + A_{50} = 72\pi + 40\pi + 9\pi = 121\pi$

Step 3: Calculate the Probability

The total area of the target: $A_{\text{total}} = A_1 = 361\pi$

The probability that Jake will earn at least 30 points is the ratio of the area of the regions scoring at least 30 points to the total area of the target: $P(\text{at least 30 points}) = \frac{A_{\geq 30}}{A_{\text{total}}} = \frac{121\pi}{361\pi} = \frac{121}{361} = \frac{1}{3} \approx 0.333$

So, the probability that Jake will earn at least 30 points on a throw, given that he hits the target, is approximately $\frac{1}{3}$ or 33.3%.

Summary

• Jake should evaluate if a 33.3% chance of winning is worth the risk compared to accepting the tie.
• Jake has a one-third probability of earning at least 30 points on his next throw, which is a key factor in his decision-making.

Would you like more details or have any questions?

Here are some related questions to explore further:

1. What is the probability that Jake will earn exactly 30 points?
2. How would the probability change if the radius of the 50-point region increased?
3. What is the probability that Jake will earn less than 30 points?
4. How does the width of each region affect the overall probability distribution?
5. If Jake's dart-throwing accuracy improves, how does that affect his decision?
6. What is the expected value of the points Jake will score in a single throw?
7. How does the number of regions affect the probability calculations?
8. How would introducing additional scoring regions (e.g., 60, 70 points) change the probabilities?

Tip: When making decisions involving probability, always weigh the risks and benefits based on the likelihood of different outcomes.