Final answer:
The mean number of cases of the rare tumor in a group of 19,509 children is 0.215. To calculate probabilities for specific case scenarios such as having 0 or 1 case, or more than one case, a Poisson distribution is used. The appearance of four cases in a town is analyzed against the expected mean to assess whether it's due to random chance.
Step-by-step explanation:
The rare form of malignant tumor that occurs in 11 children out of a million can be analyzed statistically to determine the number of expected cases in a given population, as well as the probabilities of different numbers of cases occurring.
Mean Number of Cases
To calculate the mean number of cases (mean) in groups of 19,509 children, we use the probability of the event (0.000011) multiplied by the number of trials (19,509 children):
Mean = Probability × Number of Children
Mean = 0.000011 × 19,509
Mean = 0.214599
We can round this to three decimal places as needed, resulting in a mean of 0.215 cases.
Probability for 0 or 1 Case
Using a Poisson distribution with the unrounded mean, we can calculate the probabilities for 0 or 1 tumor case. However, the necessary computation details and formula are not provided in this summary.
Probability of More Than One Case
The probability of more than one case can also be found using the Poisson distribution.
Cluster of Four Cases Analysis
The cluster of four cases would be considered against the expected mean (0.215) to determine if it is attributable to random chance.