Statistics and Probability Answers

6. A certain company has two car assembly plants, A and B. Plant A produces twice as many cars as plant B. Plant A uses engines and transmissions from a subsidiary plant which produces 10% defective engines and 2% defective transmissions. Plant B uses engines and transmissions from another source where 8% of the engines and 4% of the transmissions are defective. Car transmissions and engines at each plant are installed independently.

(a) What is the probability that a car chosen at random will have a good engine?
(b) What is the probability that a car from plant A has a defective engine, or a defective transmission, or both?
(c) What is the probability that a car which has a good transmission and a defective engine was assembled at plant B?

5. A large industrial firm uses three local motels to provide overnight accommodations for its clients. From past experience it is known that 20% of the clients are assigned rooms at the Ramada Inn, 50% at the Sheraton, and 30% at the Lakeview Motor Lodge. If the plumbing is faulty in 5% of the rooms of the Ramada Inn, in 4% of the rooms at the Sheraton, and in 8% of the rooms at the Lakeview Motor Lodge, what is the chance that
Total Probability & Bayes' Rule - Answer must be in PERCENTAGES.

(a) a client will be assigned a room with no faulty plumbing
(b) a person with a room have faulty plumbing was assigned accommodations at the Sheraton

4. A manufacturer of a flu vaccine is concerned about the quality of its flu serum. Batches of serum are processed by three different departments having rejection rates of 0.10, 0.08, and 0.12, respectively. The inspections by the three departments are sequential and independent.
Conditional Probability, Independence, Product Rule - Answers must be in DECIMAL form.

(a) What is the probability that a batch of serum survives the first departmental inspections but is rejected by the second department?
(b) What is the probability that a batch of serum is rejected by the third department?

3. Factory workers are constantly encouraged to practice zero tolerance when it comes to accidents in factories. Accidents can occur due to carelessness or so-called human error. In addition, the worker's shift, 7:00 AM - 3:00 PM (day shift), 3:00 PM - 11:00 PM (evening shift), or 11:00 PM - 7:00 AM (graveyard shift), may be a factor. During the last year, 300 accidents have occurred. The percentages of the accidents for the condition combinations are as shown in the table below. If an accident report is selected randomly from the 300 reports,
SHIFT UNSAFE HUMAN
CONDITION ERROR
DAY 5% 32%
EVENING 6% 25%
GRAVEYARD 2% 30%
(a) what is the chance that the accident occurred on the evening shift?
(b) what is the chance that the accident occurred on either the day or the graveyard shift?
(c) what is the chance that the accident did not happen in the graveyard shift?

2. The probabilities that a service station will pump gas into 0, 1, 2, 3, 4, or 5 or more cars during a certain 30-minute period are 0.03, 0.18, 0.24, 0.28, 0.10, and 0.17, respectively. Find the probability that in this 30-minute period

(a) more than 3 cars receive gas
(b) 2 or 3 cars receive gas
(c) 7 cars receive gas

1. A box containing 6 black balls, 4 green balls, and 3 red balls. 3 balls are drawn in succession, wherein each ball being replaced in the box before the next draw is made. What is the probability that

(a) all 3 are the same color?
(b) each color is represented?

You have a deck of 52 playing cards.
(i) How many different 5 card hands can be dealt?
(ii) What is the probability that a hand of 5 dealt randomly contains 3 aces?
(iii) What is the probability that a hand of 5 dealt randomly will have 5 cards of the same

At the end of a production process, 500 electrical components pass through three quality
checks: P, Q and R. It is found that 38 components fail check P, 29 fail Q, 30 fail R, 7 fail P
and Q, 5 fail Q and R, 8 fail R and P, and 3 fail all checks. Determine how many components:
(a) pass all checks,
(b) fail exactly one check,
(c) fail exactly two checks.

A box contains 22 balls of which 7 are red, 9 are white, and 6 are black. Four balls are drawn
at random from the box without replacement. Find the probability that
(a) three red balls and one white ball are drawn,
(b) the four balls are all red,
(c) the balls are all of the same colour,
(d) there is at least one ball of each colour.

You and a friend play a spin-the-wheel game. You have a wheel with 20 numbers (1-20) which can be spun, and with each spin, one of the numbers is randomly selected. Your friend offers you a bet: If the numbers on the next three spins contain at least two odd numbers, then they will pay you €15, otherwise you will pay them €15. To the nearest euro how, many euro would you expect to win/lose if you gambled 92 times on this bet?