Problem 199 - An Introduction to Mathematical Statistics and Its Applications

Question

Marksmanship competition at a certain level requires each contestant to take ten shots with each of two different handguns. Final scores are computed by taking a weighted average of four times the number of bull's-eyes made with the first gun plus six times the number gotten with the second. If Cathie has a \(30 \%\) chance of hitting the bull's-eye with each shot from the first gun and a \(40 \%\) chance with each shot from the second gun, what is her expected score?

Step-by-Step Solution

Answer

In conclusion, Cathie's expected final score in the marksmanship competition is 36.

Step 1: Calculate the expected number of bull's-eyes with the first gun

Firstly, calculate the expected number of bull's-eyes that Cathie would get with the first gun. This is done by multiplying the number of shots (10) by the probability of success (0.30), using the formula for expected value for a binomial distribution. Therefore, the expected bull's-eyes with the first gun is \(10 * 0.30 = 3\).

Step 2: Calculate the expected number of bull's-eyes with the second gun

Next, calculate the expected number of bull's-eyes that Cathie would get with the second gun. Similarly, this is done by multiplying the number of shots (10) by the probability of success (0.40). Therefore, the expected bull's-eyes with the second gun is \(10 * 0.40 = 4\).

Step 3: Calculate the final score

Finally, the score is computed by taking a weighted average of four times the number of bull's-eyes made with the first gun and six times the number gotten with the second. So, the final expected score is calculated as desired: \(4*3 + 6*4 = 36\).

Key Concepts

Binomial Distribution

Imagine you are taking part in a contest where you shoot at a target. Each time you shoot, you can hit the bull's-eye, or miss it. In such a scenario, we can use a mathematical tool called the binomial distribution to help us figure out the expected outcomes.

Binomial distribution is used when there are a fixed number of independent trials (like shooting a certain number of times), each with two possible outcomes (hit or miss). If you know the probability of hitting the target with each shot, you can calculate the expected number of times you'll hit the target in a given number of shots.

For Cathie, when she uses the first gun, she shoots 10 times. She has a 30% chance with each shot. Using the formula for expected value in a binomial distribution: Expected Value = Number of trials × Probability of success, we calculate that she is expected to hit the bull’s-eye 3 times with this gun.

Probability of Success

Whenever you're faced with making predictions or calculating outcomes involving chance, understanding the probability of success is crucial.

Probability of success in an event is the likelihood of a desired outcome occurring in a single trial. In the case of shooting and hitting a bull's-eye, this probability tells us how likely Cathie is to hit the target with each shot she takes.

For her first gun, Cathie's probability of hitting the bull's-eye is 0.30 (or 30%), meaning out of 10 shots, on average you would expect her to hit the target 3 times. Similarly, with her second gun, the probability of success is 0.40 (or 40%), resulting in an expected value of 4 hits after 10 attempts.

Weighted Average

A weighted average is handy when different values in a dataset contribute unequally to the final result.

In Cathie's case, she scores differently with the results from each gun. The first gun's score gets multiplied by 4, and the second's by 6. These are the weights given to the results of each gun and this affects the final expected score.

To calculate the weighted average, you multiply each expected outcome by its weight and sum them up. Here, it results in this calculation:

• First gun: 3 expected hits × 4 = 12
• Second gun: 4 expected hits × 6 = 24

Adding these gives Cathie’s final score of 36. The weights show which outcomes are more valued in calculating her score.