Question

Both Alice and Bob toss a fair coin three times. The probability that Alice records a different numbers of heads than Bob is given by A/B, where A and B are relatively prime integers (greatest common divisor is 1). Find A + B.

Answer 1

Sample space of tossing a fair coin 3 times is

{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

Explanation: Getting 3 heads, 2 heads first time and tails 3rd time and so on

There can be 3 heads , 2 heads ,1 head or 0 heads out of 8 outcomes. The probabilities for Alice and Bob will be the same for same number of heads individually. The probability will be therefore

No. of heads	Probability
3	1/8
2	3/8
1	3/8
0	1/8

So if the recording are different the possible pairs of recordings are

Alice	Bob	Probability P(Alice ) * P(Bob)
3	2, 1, 0	1/8 * (3/8 + 3/8 + 1/8) = 7/64
2	3, 1, 0	3/8 * (1/8 + 3/8 + 1/8) = 15/64
1	3, 2, 0	3/8 * (1/8 + 3/8 + 1/8) = 15/64
0	3, 2, 1	1/8 * (1/8 + 3/8 + 3/8) = 7/64

Explanation: We multiply the possibility of Alice and Bob since the events are simultaneous (intersection).

  We add the possibility of Bob because we don't which number he would get so we consider all possibilities (union)

The total probability adds upto = (7 + 15 + 15 + 7 ) /64 ...again we add since we don't which pair might be recorded.

= 44/64

= 11/16 ........divided by

Since this can't be reduced further we conclude that 11 and 16 are relatively prime numbers.

A = 11 and B = 16

Their sum = 27

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