Three heads are better than one!

Probability Level 3

You have a biased coin that lands on heads 75% of the time.

You flip it five times.

What is the probability that at some point it lands on heads at least 3 times in a row?


The answer is 0.633.

Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Nov 18, 2018

To a avoid double counting, you have the following 8 ways you can yield the desired result (where X means either heads or tails):

1) HHHXX (4)

2) THHHX (2)

3) XTHHH (2)

  • P ( 1 ) = ( 3 4 ) 3 P(1) = (\frac{3}{4})^3
  • P ( 2 ) = P ( 3 ) = 1 4 ⋅ ( 3 4 ) 3 P(2) = P(3) = \frac{1}{4} \cdot (\frac{3}{4})^3

P = P ( 1 ) + P ( 2 ) + P ( 3 ) = 0.633 P = P(1) + P(2) + P(3) = 0.633

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