Let's go through each part of the problem.

Part (a): Finding the Mean Number of Cases

To find the mean number of cases, we can use the formula for the expected value $\mu = n \times p$, where:

• $n = 15,659$ (the number of children in the town)
• $p = 0.000011$ (the probability of one child having the tumor)

$\mu = n \times p = 15,659 \times 0.000011$

Calculating this:

$\mu = 0.172249$

So, the mean number of cases in groups of 15,659 children is approximately 0.172.

Part (b): Probability that the Number of Tumor Cases is 0 or 1

With the mean $\mu = 0.172249$, we can use a Poisson distribution to approximate the probabilities, since the probability of occurrence is very low and the mean is small.

The Poisson probability formula is: $P(X = k) = \frac{e^{-\mu} \mu^k}{k!}$ where $X$ is the number of cases and $\mu = 0.172249$.

Probability of 0 cases: $P(X = 0) = \frac{e^{-0.172249} \cdot 0.172249^0}{0!} = e^{-0.172249} \approx 0.8419$

Probability of 1 case: $P(X = 1) = \frac{e^{-0.172249} \cdot 0.172249^1}{1!} \approx 0.1451$

Adding these probabilities gives us the probability of 0 or 1 cases: $P(X = 0 \text{ or } 1) = P(X = 0) + P(X = 1) \approx 0.8419 + 0.1451 = 0.987$

So, the probability of observing 0 or 1 cases is approximately 0.987.

Part (c): Probability of More than One Case

To find the probability of more than one case, we can use the complement of the probability of 0 or 1 cases.

$P(X > 1) = 1 - P(X = 0 \text{ or } 1) = 1 - 0.987 = 0.013$

Thus, the probability of more than one case is approximately 0.013.

Part (d): Analyzing the Cluster of Four Cases

Given that the probability of observing more than one case is only 0.013 (or 1.3%), the occurrence of four cases is highly unlikely if the tumor occurrences are purely random. Therefore, the cluster of four cases does not appear to be attributable to random chance alone, suggesting that there might be an external factor or cause contributing to the unusually high occurrence in this town.

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Would you like more details on any step or have any other questions?

Related Questions

1. How is the Poisson distribution useful in analyzing rare events?
2. Could the result in part (d) justify further epidemiological study in this town?
3. What other factors might explain an unusually high rate of rare diseases in a specific location?
4. How would a larger population size affect the mean and probabilities?
5. What is the relationship between probability and statistical significance in public health studies?

Tip:

For rare-event probability problems, the Poisson distribution provides a useful approximation when the sample size is large, and individual probabilities are small.