Question

Required:

Show your complete solution· Write your formula (explicitly) first before actually solving the problem *Clearly define your random variables or events.

1. A company uses four machines to produce widgets. Machine A produces 30% of the widgets of which 3% are defective. Machine B produces 25% of the widgets of which 5% are defective. Machine C produces 10% of the widgets of which 4% are defective. Machine D produces 35% of the widgets of which 2% are defective. All produced widgets are places in a storage area. If one widget if selected at random, a. What is the probability that it is defective? b. What is the probability that it comes from Machine A or B and it is not defective? c. What is the probability that it is not defective and produced by Machine D?

2. An HR manager was requested to fill in 2 data science positions. From 100 candidates only 10 candidates passed the qualifying exam. Among the 10 candidates, 3 are women and 7 are men. If every candidate is equally likely to be chosen, what is the probability that no women will be hired?

3. As a payment for your help, you have been given a choice of either a flat payment of $5, or a chance of randomly drawing a bill from a box. The box contains one $100 bill, two $20 bills, seven $10 bills, ten $5 bills, and thirty $1 bills. What choice will you take, the flat payment or draw from the box? Show your solution that proves your decision. (ASSIGN P(formula)

4. scientist you want to prove/disprove this claim by being data driven. You randomly select 15 boxes of cereal produced by this manufacturer and find that 5 are underweight. What is the probability of selecting 5 or more underweight boxes of cereal in a sample of 15 if the percentage of underweight boxes is what the manufacturer claims? Hint: Let X represents the number of boxes (out of 15) that are underweight.

5. Approximately 12% of the U.S. population is composed of African-Americans. Assuming that the same percentage is true for phone plan ownership, what is the probability that when 25 phone numbers are selected at random for a small survey, 5 of the phone numbers belong to an African-American family?

6. The mean number of patients arriving at the emergency room of City Hospital on Saturday nights between 10:00 and 12:00 is 6.5. Assuming that the patients arrive randomly and independently, a. What is the probability that on a given Saturday night, 5 or fewer patients arrive at the emergency room between 10:00 and 12:00? b. What is the probability that on a given Saturday night, 2 or fewer patients arrive at the emergency room between 11:00 and 12:00?

Answer 1

Given the item is defective, probability of it being produced by B=0.26

Explanation:

Prior information

Probability of an item produced by Machine A P(A)=50%=0.5
Probability of an item produced by Machine B P(B)=30%=0.3
Probability of an item produced by Machine C P(C)=20%=0.2

On the basis of additional information
Given the item produced by A, probability of it being defective P(D/A)=3%=0.03
Given the item produced by B, probability of it being defective P(D/B)=2%=0.02
Given the item produced by C, probability of it being defective P(D/C)=1%=0.01

Given the item is defective, probability of it being produced by B

P(BD)=P(B)×P(DB)/[P(B)×P(DB)]+[P(A)×P(DA)]+[P(C)×P(DC)]

=0.3×0.02[0.3×0.02]+[0.5×0.03]+[0.2×0.01]
0.0060.006+0.015+0.002=0.0060.023=0.26

Given the item is defective, probability of it being produced by B=0.26