John throws a coin until a series of three consecutive heads

Question

If John throws a coin until a series of three consecutive heads or three consecutive tails appears, what is the probability that the game will end on the fourth throw?

A.  \frac{1}{16} 
B.  \frac{1}{8} 
C.  \frac{3}{16} 
D.  \frac{1}{4} 
E.  \frac{3}{8}

Bunuel · Accepted Answer

Official Solution:

If John throws a coin until a series of three consecutive heads or three consecutive tails appears, what is the probability that the game will end on the fourth throw?

A.  \frac{1}{16} 
B.  \frac{1}{8} 
C.  \frac{3}{16} 
D.  \frac{1}{4} 
E.  \frac{3}{8}

Calculate the probability that either of the two sequences,  THHH  or  HTTT , will occur. The probability of  THHH  occurring =  (\frac{1}{2})^4 = \frac{1}{16} . The probability of  HTTT  occurring =  (\frac{1}{2})^4 = \frac{1}{16} . The probability of either  THHH  or  HTTT  occurring =  \frac{1}{16} + \frac{1}{16} = \frac{2}{16} = \frac{1}{8} .

Answer: B

mikemcgarry · Answer

Great question! I'm happy to help. :-)

Let's consider the routes that would lead to ending the game on the fourth toss. In order to be a win at that point, tosses #2 & #3 & #4 would all have to be the same, either H or T, and in order for the game NOT to end on the third toss, the first toss would have to be different from the other four. Thus, the only two possible sequences are H-T-T-T ----> P = (1/2)^4 = 1/16 T-H-H-H ----> P = (1/2)^4 = 1/16 Either one of these would satisfy the condition, so we could have one or the other. OR means add in probability. 1/16 + 1/16 = 2/16 = 1/8 Answer (B) . Incidentally, the GMAT would NEVER give the unsimplified fraction 2/16 --- it would always give the simplified form 1/8. Does all this make sense? Mike :-)

shrikar23 · Answer

ans is B.
since the coin is tossed 4 times there can be 16 total outcomes.
there are only 2 outcomes that satisfy the condition.
HTTT and THHH.

SchruteDwight · Answer

1) THHH
2) HHHT
3) TTTH
4) HTTT

4 favourable outcomes (notice that HHHH or TTTT is not possible) out of 2*2*2*2 total outcomes
= 2/16=1/8

spy2206 · Answer

hi  Bunuel ,  mikemcgarry , since there are four favourable outcomes- HHHT, THHH, TTTH, HTTT- isn't it supposed to be 4/16 ?? I did not understand the above solution please help

Bunuel · Answer

HHHT and TTTH does not work because in these cases the game ends on third throw not on the fourth.

spy2206 · Answer

OHHH RIGHTT. THANK YOUUU!!!!!

Saimithlesh · Answer

Hello is solving the question this way also correct?
1st toss- can be either head or tails P=1
2nd toss- can only be either heads or tails P=1/2
3rd toss- can only be either heads or tails P=1/2
4th toss- can only be either heads or tails P=1/2
So overall probability is 
1*1/2*1/2*1/2= 1/8

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John throws a coin until a series of three consecutive heads

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John throws a coin until a series of three consecutive heads

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