A coin is biased so that a head is twice as likely to occur as a tail. This coin

Question

A coin is biased so that a head is twice as likely to occur as a tail. This coin is tossed by three people in a two-step gambling game. In step 1 the person whose toss is not a match to either of the other two people is "odd man out. Only the remaining two players whose coin match go on to step 2 to resolve the ultimate winner. (a) What is the probability that a player will advance to step 2 after the first toss? (b) What is the probability that a player will be out after the first toss? (c) What is the probability that no player will be out after the first toss?

Explanation / Answer

There are 9 ways in the 1st step . Let three people be X,Y and Z .
X Y Z probability
H H H 8/27
H H T 4/27 (Z becomes out)
H T H 4/27 (Y becomes out)
H T T 2/27 (X becomes out)
T H H 4/27 (X becomes out)
T H T 2/27 (Y becomes out)
T T H 2/27 (Z becomes out)
T T T 1/27
"a player" can be any one of X,Y and Z ,
(a) the probability that a player will advance to step 2 after the first toss : 1 - 6/27 = 21/27 = 7/9
(b) the probability that a player will be out after the first toss : 6/27 = 2/9
(c) the probability that no player will be out after the first toss : 9/27 = 1/3

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