An arcade booth at a county fair has a person pick a coin from two possible coin

Question

An arcade booth at a county fair has a person pick a coin from two possible coins available and then toss it. If the coin chosen lands on heads, the person gets a prize. One coin is a fair coin and one coin is a biased coin (unfair) with only a 28% chance of getting a head. Assuming equally likely probability of picking either coin, what is the probability that the fair coin is the one chosen, given that the chosen coin lands on heads?

a) 0.6410

b) 0.4717

c) 0.7800

d) 0.3900

e) 0.0175

King Mattress purchases Sleep-n-Air mattresses from three different distributors. The probability of getting a defective mattress from Distributor A, B, or C is 0.42, 0.15, and 0.23, respectively. Assume an equal probability of making a purchase from each distributor A, B, or C. If King Mattress sells a defective mattress, what is the probability that it came from Distributor A?

a) 0.5250

b) 0.0714

c) 0.1875

d) 0.2875

e) 0.2000

An experiment consists of choosing an urn with the following probabilities that Urn 1, Urn 2, or Urn 3 will be chosen: 1/2, 1/4, and 1/4, respectively. Urn 1 contains 6 brown marbles and 8 clear marbles. Urn 2 contains 15 brown marbles, 12 clear marbles and 11 red marbles. Urn 3 contains 13 brown marbles, 7 clear marbles and 5 red marbles.   

A marble is then chosen from the chosen urn. What is the probability that Urn 3 was chosen, given that the marble chosen was clear?

a) 0.2399

b) 0.0933

c) 0.2800

d) 0.0700

e) 0.1610

Explanation / Answer

Let F shows the event that a fair coin is chosen and B shows the event that a biased coin is chosen. So we have

P(F) = P(B) =0.5

Now H shows the event that coin lands on head. So we have

P(H | F) = 0.5, P(H |B) = 0.28

By the law of total probability, the probability that coin land on head is

P(H) = P(H|F)P(F) + P(H|B) P(B) = 0.5 * 0.5 + 0.28 *0.5 = 0.39

By the Baye's theorem, the requried probability is

P(F |H) = [ P(H|F) P(F) ] / P(H) = [0.5 * 0.5] / 0.39 = 0.6410

Correct option is a.

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