A state highway department has contracted for the delivery o

A state highway department has contracted for the delivery of sand, gravel, and
cement at a construction site. Because of other work commitments and labor
force problems, contracting firms cannot always deliver items on the agreed
delivery date. From past experience, the probabilities that sand, gravel, and
cement will be delivered on the promised delivery dates by the contracting firms
are 0.3, 0.6, and 0.8, respectively. Assume that the delivery or nondelivery of
one material is independent of another.
(a) Find the probability that all three materials will be delivered on time.
(b) Find the probability that none of the three materials will be delivered on
time.
(c) Find the probability that at least one of the materials will be delivered on
time.

Solution

A state highway department has contracted for the delivery of sand, gravel, and
cement at a construction site. Because of other work commitments and labor
force problems, contracting firms cannot always deliver items on the agreed
delivery date. From past experience, the probabilities that sand, gravel, and
cement will be delivered on the promised delivery dates by the contracting firms
are 0.3, 0.6, and 0.8, respectively. Assume that the delivery or nondelivery of
one material is independent of another.
(a) Find the probability that all three materials will be delivered on time.
(b) Find the probability that none of the three materials will be delivered on
time.
(c) Find the probability that at least one of the materials will be delivered on
time.

Solution : P(all delivered) = 0.3 * 0.6 * 0.8 = 0.144

b) P(none delivered) = (1 - 0.3)(1 - 0.6)(1 - 0.8) = 0.056

c) P(atleast one delivered) = 1 - P(none delivered) = 1 - 0.056 = 0.944