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Chapter 8: Problem 61

Must the expected number of times you hit a bull's-eye after 50 attempts always be a whole number? Explain.

Short Answer

Expert verified
The expected number of times a bull's-eye is hit after 50 attempts does not always need to be a whole number. It depends on the probability (p) of hitting a bull's-eye in a single attempt. The expected value (E) is calculated using the formula \(E = 50 * p\), and since 'p' can be any value between 0 and 1, the expected value 'E' can be any real number between 0 and 50, not just whole numbers.

Step by step solution

01

1. Understanding Expected Value

Expected value is the average number of successes we can expect in a large number of trials. It is calculated by multiplying the probability of success in each trial by the total number of trials. In this problem, we are interested in the expected value of hitting a bull's-eye after 50 attempts.
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2. Finding the Probability of Hitting a Bull's-Eye

In order to calculate the expected value, we first need to know the probability of hitting a bull's-eye in a single attempt. The probability can vary depending on several factors, such as the person's skill or the distance between the person and the target. The exercise does not provide this information, but we can use the generic probability value 'p', where 0 <= p <= 1, to represent the probability of hitting a bull's-eye.
03

3. Calculating the Expected Value

Now that we have the probability 'p', we can use it to find the expected value of hitting a bull's-eye after 50 attempts. The formula for expected value (E) is: \[E = n * p\] where n is the number of trials (50 attempts) and p is the probability of success (hitting a bull's-eye) in each trial. In our case, it would be: \[E = 50 * p\]
04

4. Analyzing the Result

In the formula \(E = 50 * p\), 'p' can be any value between 0 and 1, and it does not necessarily have to be a fraction with a denominator that divides 50 (e.g., 1/2, 1/4, 1/10, and so on). Consequently, the expected value 'E' can be any real number between 0 and 50, not just whole numbers. For example, if the probability of hitting a bull's-eye is 0.1 (1 in 10 attempts), then the expected value after 50 attempts would be: \[E = 50 * 0.1 = 5\] However, if the probability is 1/3, then the expected value would be: \[E = 50 * \frac{1}{3} \approx 16.67\]
05

5. Conclusion

The expected number of times a bull's-eye is hit after 50 attempts does not always need to be a whole number. It depends on the probability of hitting a bull's-eye in a single attempt, which can result in an expected value that is a real number rather than just a whole number.

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