
A rare form of malignant tumor occurs in 11 children in a​ million, so its probability is 0.000011. Four cases of this tumor occurred in a certain​ town, which had 15,659 children.

**a.** Assuming that this tumor occurs as​ usual, find the mean number of cases in groups of 15,659 children.

**b.** Using the unrounded mean from part ​(**a**​), find the probability that the number of tumor cases in a group of 15,659 children is 0 or 1.

**c.** What is the probability of more than one​ case?

**d.** Does the cluster of four cases appear to be attributable to random​ chance? Why or why​ not?

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.

A rare form of malignant tumor occurs in 11 children in a million, so its probability is 0.000011. Four cases of this tumor occurred in a certain town, which had 15,659 children.

**a.** Assuming that this tumor occurs as usual, find the mean number of cases in groups of 15,659 children.

**b.** Using the unrounded mean from part (a), find the probability that the number of tumor cases in a group of 15,659 children is 0 or 1.

**c.** What is the probability of more than one case?

**d.** Does the cluster of four cases appear to be attributable to random chance? Why or why not?

Explore a probability problem involving a rare form of malignant tumor occurring in children. This problem calculates the expected number of cases, probabilities for 0 or 1 occurrences, and more than one case using Poisson distribution. Suitable for high school students, Grades 11-12.

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