The Probability the number of flips of a fair coin to achieve 3 heads is mod 3Compute the probability that the number of flips of a fair coin needed in order to achieve a total of exactly 3 Heads will be a multiple of 3.

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Catherine Kirby · Accepted Answer

Step 1Keeping flipping a coin until a head is achieved. We have ...The probability that exactly 1 heads will be achieved in 1(mod3) flips is       4    7  .The probability that exactly 1 heads will be achieved in 2(mod3) flips is       2    7  .The probability that exactly 1 heads will be achieved in 0(mod3) flips is       1    7  .Step 2Now consider achieveing 3 heads, the first after a flips, the second after a further b flips &amp;amp; the third after a further c flips. We require   a  +  b  +  c  ≡  0    (  mod    3  ). And a,b,c can be chosen in the following ways...Now   1  +  1  +  1 (In one way) or   2  +  2  +  2 (In one way) or   0  +  0  +  0 (In one way) or   0  +  1  +  2 (in 6 ways) so ...                    p        =                              (                          4              7                        )                    3                +                              (                          2              7                        )                    3                +                              (                          1              7                        )                    3                +        6        ×                  1          7                          2          7                          4          7

The Probability the number of flips of a fair coin to achieve 3 heads is mod3. Compute the probability that the number of flips of a fair coin needed in order to achieve a total of exactly 3 Heads will be a multiple of 3.

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