Question

1) A door-to-door salesman expects to make a sale 26% of the time when starting the day. But making a sale increases his enthusiasm so much that the probability of a sale to the next customer is 0.36. If he makes no sale, the probability for a sale to the next customer stays at 0.26. What is the probability that he will make at least two sales with his first three visits?

2)Two machines turn out all the products in a factory, with the first machine producing 50% of the product and the second 50%. The first machine produces defective products 2% of the time and the second machine 7% of the time.

(a) What is the probability that a defective part is produced at this factory given that it was made on the first machine?


(b) What is the probability that a defective part is produced at this factory?

3)Dystopia county has three bridges. In the next year, the Elder bridge has a 11% chance of collapse, the Younger bridge has a 2% chance of collapse, and the Ancient bridge has a 22% chance of collapse. What is the probability that exactly one of these bridges will collapse in the next year? (Round your final answer to four decimal places. Do not round intermediate calculations.)

Answer 1

Solution:

1) Let S denotes sale and F denotes no sale.
We are given that probability of sale is 0.26 and probability of consecutive sale is 0.36.
Therefore, probability of no sale = 1 - 0.26 = 0.74
probability of no sale just after sale = 1 - 0.36 = 0.64
He will make at least two sales with his first three visits, if he has outcomes:
SSS, SSF, SFS, FSS
P(at least two sales with his first three visits) = P(SSS) + P(SSF) + P(SFS) + P(FSS)
= (0.26)(0.36)(0.36) + (0.26)(0.36)(0.64) + (0.26)(0.64)(0.26) + (0.74)(0.26)(0.36)
= 0.033696 + 0.059904 + 0.043264 + 0.069264
= 0.206128
--------------------------------------------------------------------------------------------------------------------------
2)
a) The probability that a defective part is produced at this factory given that it was made on the first machine
P = 0.02
b) The probability that a defective part is produced at this factory
P = 0.50*0.02 + 0.50*0.07 = 0.045
------------------------------------------------------------------------------------------------------------------------------
3) The required probability = 0.11*(1-0.02)*(1-0.22)+ 0.02*(1-0.22)*(1-0.11) + 0.22*(1-0.02)*(1-0.11)
= 0.2898