Question

The number of knots in a piece of lumber is a random variable, X. Suppose that X has a Poisson distribution with E(X) = 4.
(a) If four independent pieces of lumber are examined, what is the probability that there is exactly one lumber has no knots?
(b) If we consider 100 pieces of lumber, write down the exact expression for the probability that the total number of knots is at least 450?
(c) Find a normal approximation to the probability in (b).

Thank you!

Answer 1

a) Probability of no knot in a piece of lumber is computed as:
P(X = 0) = e^-4 = 0.0183

Therefore the probability that there is exactly one lumber has no knots is computed here as:

Therefore 0.0693 is the required probability here.

b) for 100 pieces of lumber, the distribution of the number of knots in 100 pieces is modelled here as:

The probability now is computed here as:

P(Y >= 450) = 1 - P(X <= 449)

This is computed in EXCEL as:
=1-poisson.dist(449,400,TRUE)

0.0075 is the output here.

Therefore 0.0075 is the required probability here.

c) The given distribution in b) is approximated to a normal distribution here as:

The probability required here is:
P(Y >= 450)

Applying the continuity correction, we have here:
P( Y > 449.5)

Converting it to a standard normal variable, we have here:

Getting it from the standard normal tables, we get here:

Therefore 0.0067 is the required approximated probability here.