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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2022-08-21

The Probability the number of flips of a fair coin to achieve 3 heads is mod 3
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.

Answer & Explanation

Catherine Kirby

Catherine Kirby

Beginner2022-08-22Added 10 answers

Step 1
Keeping 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 2
Now consider achieveing 3 heads, the first after a flips, the second after a further b flips & 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

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