We are tasked with finding the approximate probability that the archer hits a bull's eye at least 26 times, using the normal approximation to the binomial distribution.

Step 1: Define the binomial random variable

Let $X$ represent the number of bull's eyes the archer hits. Since the number of attempts is 81 and the probability of hitting a bull's eye in a single attempt is $\frac{1}{3}$: $X \sim \text{Binomial}(n = 81, p = \frac{1}{3})$

The mean ($\mu$) and standard deviation ($\sigma$) of a binomial random variable are: $\mu = n \cdot p = 81 \cdot \frac{1}{3} = 27$ $\sigma = \sqrt{n \cdot p \cdot (1-p)} = \sqrt{81 \cdot \frac{1}{3} \cdot \frac{2}{3}} = \sqrt{18} \approx 4.24$

Step 2: Apply the normal approximation

Using the normal approximation, $X$ can be approximated as: $X \sim N(\mu = 27, \sigma = 4.24)$

Step 3: Find $P(X \geq 26)$

For the normal approximation, apply the continuity correction. To approximate $P(X \geq 26)$, we calculate $P(X \geq 25.5)$: $P(X \geq 26) \approx P(Z \geq \frac{25.5 - \mu}{\sigma})$

Substitute the values of $\mu$ and $\sigma$: $Z = \frac{25.5 - 27}{4.24} = \frac{-1.5}{4.24} \approx -0.35$

Using the standard normal distribution table or a calculator, find $P(Z \geq -0.35)$. From symmetry of the normal distribution: $P(Z \geq -0.35) = 1 - P(Z < -0.35) \approx 1 - 0.3632 = 0.6368$

Final Answer:

The approximate probability that the archer hits the bull's eye at least 26 times is: $P(X \geq 26) \approx 0.6368$

Would you like further details or a step-by-step derivation of the continuity correction?

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5 Related Questions:

1. What if the probability of hitting the bull's eye changes to $\frac{1}{4}$? How would that affect the result?
2. How does the continuity correction improve the accuracy of the normal approximation?
3. What is the probability that the archer hits the bull's eye fewer than 20 times?
4. How does the result change if the archer shoots 100 times instead of 81?
5. Can you explain why the normal distribution can approximate the binomial distribution in this case?

Tip:

When using the normal approximation to the binomial distribution, always check if $n \cdot p \geq 5$ and $n \cdot (1 - p) \geq 5$ to ensure the approximation is valid.