what is the probability that a sequence of five flips of a fair coin will not land heads up twice in a row?

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20-12-2022
Mathematics

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what is the probability that a sequence of five flips of a fair coin will not land heads up twice in a row?

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debjitbhowal4568 debjitbhowal4568 · Answer 1 · 2022-12-30T06:24:38-05:00

The probability that a sequence of five flips of a fair coin will not land heads up twice in a row is 13/32.

My strategy is that if we cannot have two heads in a row, we also cannot have three, four, or five heads in a row. Noting the similarity between these two occurrences:

{no HH}={not (exactly HH or not exactly HHH or exactly HHHH)}={not exactly HH and not exactly HHH and not exactly HHHH}.

These can then be expressed as complementary probabilities as follows:

P(no HH)=P(not exactly HH) ×P(not exactly HHH)× P(not exactly HHHH)× P(not exactly HHHHH)=1−P(exactly HH)×1−P(exactly HHH)×1−P(exactly HHHH)×1−P(exactly HHHHH)

We can easily compute these probabilities:

P(exactly HH)P(exactly HHH)P(exactly HHHH)P(exactly HHHHH)=425=325=225=125

The likelihood that there will be no double heads anywhere in the sequence is thus:

∏i=14(1−i25)≈0.720

Hence, A sequence of five fair coin flips has a 13/32 chance of not landing heads up twice in a row.

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