Problem 14BSC

Use the Poisson distribution to find the indicated probabilities.

Disease Cluster Neuroblastoma, a rare form of malignant tumor, occurs in 11 children in a million, so its probability is 0.000011. Four cases of neuroblastoma occurred in Oak Park, Illinois, which had 12,429 children.

a. Assuming that neuroblastoma occurs as usual, find the mean number of cases in groups of 12,429 children.

b. Using the unrounded mean from part (a), find the probability that the number of neuroblastoma cases in a group of 12,429 children is 0 or 1.

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

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

Problem 14BSC

Answer:

Step1 of 4:

We have Neuroblastoma, a rare form of malignant tumor, occurs in 11 children in a million, so its probability is 0.000011. Four cases of neuroblastoma occurred in Oak Park, Illinois, which had 12,429 children.

Consider a random variable “x” follows poisson distribution with sample size “n” and parameter “”.

i,e.X P()

Probability mass function of poisson distribution is given by

P(x) = , x = 0,1,2,...,n.

Where, e = 2.71828(Eulerian number).

Mean and variance of poisson distribution both are both given by

And .

Step2 of 4:

a).

Assuming that neuroblastoma occurs as usual,the mean number of cases in groups of 12,429 children is given by

= 12429(0.000011)

= 0.1367.

Step3 of 4:

b).

Using the unrounded mean from part (a),the probability that the number of neuroblastoma cases in a group of 12,429 children is 0 is given by P(x = 0) = ?

Consider,

P(x) =

P(x = 0) =

=

= 0.8722

Similarly,

The probability that the number of neuroblastoma cases in a group of 12,429 children is 1 is given by P(x = 1) = ?

Consider,

P(x) =

P(x = 1) =

=

= 0.1192.

Therefore,the probability that 0 or 1 case of neuroblastoma occurs in groups of 12,429 children is P(x = 0) + P(x = 1) = 0.8722 + 0.1192

= 0.9914.

Step4 of 4:

c).

The probability of more than one case of neuroblastoma is given by P(x > 1) = ?

P(x > 1) = 1 - P(x 1)

= 1 - 0.9914

= 0.0086.

Therefore,The probability of more than one case of neuroblastoma is 0.0086.

Step1 of 1:

c).

“no”,the cluster of four cases does not appear to be attributable to random chance because,the probability of more than one case extremely small so, the probability of getting as many as four cases is even smaller.