Answer :
Sure, let's break down the process to find the probability that Jake will earn at least 30 points on a throw, given that he hits the target and each region is equally likely.
### Step-by-Step Process:
1. Identify the Possible Point Values:
Determine all the distinct point values associated with hitting different regions of the target. For instance, let's say the target has regions worth 10, 20, 30, 40, and 50 points.
2. Count the Total Number of Point Values:
Count the total number of distinct point values Jake can hit. In our example, that would be 5 point values (10, 20, 30, 40, and 50).
3. Identify Point Values of Interest:
Identify and count the point values that are at least 30 points. From our example:
- 30 points
- 40 points
- 50 points
So, there are 3 point values that are at least 30 points.
4. Calculate the Probability:
The probability [tex]\(P\)[/tex] of Jake earning at least 30 points is calculated by the ratio of the number of point values that are at least 30 to the total number of point values.
[tex]\[ P(\text{at least 30 points}) = \frac{\text{Number of point values >= 30 points}}{\text{Total number of point values}} \][/tex]
5. Express the Probability:
Using the counts from the example:
- Number of point values >= 30 points = 3
- Total number of point values = 5
[tex]\[ P(\text{at least 30 points}) = \frac{3}{5} = 0.6 \][/tex]
### Conclusion:
Jake has a 0.6 or 60% probability of earning at least 30 points on a single throw.
### Decision-Making:
To decide whether Jake should accept the tie or make another throw, he should consider his probability of winning (60%) versus the certainty of a tie. This decision may also depend on how much Jake values winning compared to the risk of losing.
By following these steps, you can determine the probability Jake has for earning at least 30 points and help him make an informed decision.
### Step-by-Step Process:
1. Identify the Possible Point Values:
Determine all the distinct point values associated with hitting different regions of the target. For instance, let's say the target has regions worth 10, 20, 30, 40, and 50 points.
2. Count the Total Number of Point Values:
Count the total number of distinct point values Jake can hit. In our example, that would be 5 point values (10, 20, 30, 40, and 50).
3. Identify Point Values of Interest:
Identify and count the point values that are at least 30 points. From our example:
- 30 points
- 40 points
- 50 points
So, there are 3 point values that are at least 30 points.
4. Calculate the Probability:
The probability [tex]\(P\)[/tex] of Jake earning at least 30 points is calculated by the ratio of the number of point values that are at least 30 to the total number of point values.
[tex]\[ P(\text{at least 30 points}) = \frac{\text{Number of point values >= 30 points}}{\text{Total number of point values}} \][/tex]
5. Express the Probability:
Using the counts from the example:
- Number of point values >= 30 points = 3
- Total number of point values = 5
[tex]\[ P(\text{at least 30 points}) = \frac{3}{5} = 0.6 \][/tex]
### Conclusion:
Jake has a 0.6 or 60% probability of earning at least 30 points on a single throw.
### Decision-Making:
To decide whether Jake should accept the tie or make another throw, he should consider his probability of winning (60%) versus the certainty of a tie. This decision may also depend on how much Jake values winning compared to the risk of losing.
By following these steps, you can determine the probability Jake has for earning at least 30 points and help him make an informed decision.
