Question

Delivery time can be classified as early, on time or, late. Delivery distance can be classified as within 5 miles, between 5 and 10 miles, and over 10 miles.

From the previous records, 15% of deliveries arrive early, and 55% arrive on time. 70% of orders are within 5 miles, and 25% of orders are between 5 and 10 miles.

The probability for arriving on time if delivery distance is over 10 miles is 0. The probability of a shipment arriving on time and having a delivery distance between 5 and 10 miles is 10%. The probability for arriving early if delivery distance is within 5 miles is 20%.

Part A: What is the probability that the delivery will arrive on time if the distance is between 5 and 10 miles?

Part B: What is the probability that the delivery will arrive on time if the distance is within 5 miles?

Part C: What is the probability that the delivery will arrive late if the distance is within 5 miles?

Please answer all 3 parts thank you!

Answer 1

P(within 5 miles) = 0.70

P(between 5 and 10 miles) = 0.25

P(over 10 miles) = 0.05

P(early) = 0.15

P(on time) = 0.55

P(late) = 0.30

P(on time | over 10 miles) = 0

P(between 5 and 10 miles and on time) = 0.10

P(early | within 5 miles) = 0.20

A)

P(on time | between 5 and 10 miles)

= P(between 5 and 10 miles and on time)/P(between 5 and 10 miles)

= 0.10/0.25 = 0.40

B)

P(on time | within 5 miles)

Now, P(on time) = P(on time | within 5 miles)*P(within 5 miles) + P(on time | between 5 and 10 miles)*P(between 5 and 10 miles) + P(on time | over 10 miles)*P(over 10 miles)

-> 0.55 = P(on time | within 5 miles)*0.70 + 0.40*0.25 + 0

-> P(on time | within 5 miles) = 0.6429

C)

P(late | within 5 miles)

Now, P(early | within 5 miles) + P(on time | within 5 miles) + P(late | within 5 miles) = 1

-> P(late | within 5 miles) = 1 - 0.20 - 0.6429 = 0.1571