Question

A child counts the number of cracks in the sidewalk along the block she lives in (about 1/8 mile of sidewalk). Suppose the expected number of cracks in a block of sidewalk is 2.

a) Which distribution would be best to use to model this situation? Explain.

b) What is the probability that she observes three or more cracks?

c) What is the probability that she observed exactly two cracks in 1/2 of the block?

Answer 1

(a)

Poisson Distribution would be best to use to model this situation.

EXPLANATION:

Here number of cracks is a discrete variable. Probability of crack (p) is very small. Number of counts (n) is very large so that mean = = np = 2 is finite.

(b)

Probability Mass Function of Poisson Distribution with mean = 2 is given by:

,

                    for x = 0, 1, 2...

P(X3) = 1- [P(X=0) + P(X=1) + P(X=2)]

So,

P(X3) = 1 - 0.6767

            = 0.3233

So,

Answer is:

0.3233

(c)

mean in 1/2 of the block = = 2/2 = 1

So,

Probability Mass Function of Poisson Distribution with mean = 1 is given by:

,

                    for x = 0, 1, 2...

So,

Answer is:

0.1839