probabilty question is here?

Question

you have 3 coins, 2 fair coins and one with probability of head being 1/3. A coin was dropped and lannded head up. The probability that it is a fair coin is
(a) 1/3
(b) 1/2
(c) 3/4
(d) 3/4

Answer 1

This is a joint and conditional probability problem involving two events. The first event is that a fair coin was selected (dropped at random) from the three So first, what is the probability of dropping a fair coin? That is clearly 2/3 because two of the three coins are fair.

Then the dropped coin rolls around a bit and comes up heads for the second event. What is the probability it will land heads up? That is 1/2 on condition the coin was fair because that's the definition of a fair coin is it not? But that is only 1/3 if the dropped coin is unfair.

Thus we have the following four possible outcomes for the two events, each outcome with its own probability of occurring.

FH = P(F)*P(H|F) = 2/3 * 1/2 = 1/3***
FT = P(F)*P(T|F) = 2/3 * 1/2 = 1/3
UH = P(U) * P(H|U) = 1/3 * 1/3 = 1/9
UT = P(U)*P(T|U) = 1/3 * 2/3 = 2/9

Where F is dropping a Fair coin, U is dropping an Unfair coin, H is coming up a head, and T is coming up a tail. The sum of the probabilities above is 1.000; showing these four are all the possible outcomes for this problem.

A Fair coin has the probability P(F) = 2/3 of being dropped; so P(U) = 1 - P(F) = 1/3. If a Fair coin is dropped, the probability of a Head is P(H|F) = 1/2 and of a Tail is P(T|F) = 1/2. If an Unfair coin is dropped, the probability of a Head is P(H|U) = 1/3 and of a Tail is P(T|U) = 2/3.

The first row, for example, says a Fair (F) coin dropped and came up Head (H); where P(F) = 2/3 is the probability of a Fair coin dropping and P(H|F) = 1/2 is the probability of a Heads given the Fair coin dropped (this is a conditional probability).

In fact, as we can see above, there are two possible outcomes ending with a Heads. But only the first one, the first row, is predicated on dropping a Fair coin. So that's your answer...1/3...the one marked ***. The other possibility when a Heads comes up is that the Unfair coin dropped and the chance of that happening is 1/9, as shown by row 3 above. In other words, chances are 3:1 that if you end up with a Heads, the coin you dropped was a Fair one.

Answer 2

If each coin has the same probability of being dropped, then the probability that a dropped heads is a fair coin is 3/4.

Why are (c) and (d) the same ?

Answer 3

(a) 1/3