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 14 comma 257 children.
a. Assuming that this tumor occurs as​ usual, find the mean number of cases in groups of
14 comma 257 children.
b. Using the unrounded mean from part ​(a​), find the probability that the number of tumor cases in a group of 14 comma 257 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?

This problem deals with the probability of a rare event (the occurrence of a malignant tumor) and involves using the Poisson distribution, which is commonly applied to model the number of events that happen in a fixed interval of time or space when these events occur with a known constant mean rate.

### Part a: Mean number of cases in groups of 14,257 children

The probability of a child having this tumor is \( p = 0.000011 \).

The expected or mean number of cases \( \lambda \) in a town with 14,257 children can be calculated as:

\[
\lambda = p \times \text{number of children} = 0.000011 \times 14,257
\]

Let me calculate that.

The mean number of tumor cases in a group of 14,257 children is approximately \( \lambda = 0.157 \).

### Part b: Probability that the number of tumor cases is 0 or 1

To find the probability that the number of tumor cases is either 0 or 1, we use the Poisson probability mass function (PMF):

\[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
\]

For \( k = 0 \) and \( k = 1 \):

\[
P(X = 0) = \frac{\lambda^0 e^{-\lambda}}{0!} = e^{-\lambda}
\]
\[
P(X = 1) = \frac{\lambda^1 e^{-\lambda}}{1!} = \lambda e^{-\lambda}
\]

The total probability is:

\[
P(X = 0 \text{ or } 1) = P(X = 0) + P(X = 1)
\]

Let me calculate that using the unrounded mean of \( \lambda = 0.156827 \).

It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can explain the next steps so you can calculate the probabilities manually:

1. Use the Poisson distribution formula:
   - \( P(X = 0) = e^{-\lambda} \)
   - \( P(X = 1) = \lambda e^{-\lambda} \)
   
   Where \( \lambda = 0.156827 \), which we calculated in part (a).
   
2. Add these two probabilities to find \( P(X = 0 \text{ or } 1) \).

### Part c: Probability of more than one case

The probability of more than one case is simply:

\[
P(X > 1) = 1 - P(X = 0 \text{ or } 1)
\]

Once you have \( P(X = 0 \text{ or } 1) \), subtract that from 1 to get the probability of more than one case.

### Part d: Does the cluster of four cases appear to be attributable to random chance?

You can compare the calculated probabilities to assess this. If the probability of more than one case is extremely small, the occurrence of 4 cases would be very unlikely to be due to random chance.

---

Would you like to ask anything specific about the calculations, or should we explore more about Poisson distribution? Here are 5 follow-up questions:

1. How is the Poisson distribution used in other real-world situations?
2. Why is the Poisson distribution appropriate for this problem?
3. Can the mean \( \lambda \) change if the population size changes?
4. How can we interpret the expected number of cases in the context of real data?
5. What other statistical distributions could be used to model rare events?

**Tip**: The Poisson distribution is often applied to model rare events over a fixed space or time interval.

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 14,257 children. a. Assuming that this tumor occurs as usual, find the mean number of cases in groups of 14,257 children. b. Using the unrounded mean from part (a), find the probability that the number of tumor cases in a group of 14,257 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 how to use the Poisson distribution to model the probability of a rare malignant tumor in a population of 14,257 children. This detailed problem involves calculating the expected number of cases, finding the probability of 0 or 1 cases, and determining if a cluster of four cases is due to random chance. Suitable for Grades 11-12 and college students studying probability theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation