Jeff Vance writes:

>My wife just took a biology test where one of the questions was:
>   "What is the probability of tossing 6 heads in a row out of 20 coin
>tosses?"

Assume a fair coin, of course.

The probability of getting six heads in a row on six tosses is 1/2^6, or
1/64. The probability of not getting six heads in a row is therefore
1-1/64, or 63/64. Repeating this trial 15 times for a total of 20 tosses
gives (63/64)^15 probability of not getting six heads in a row, or
1-(63/64)^15 =~21%.

Another way to do this is to list all of the combinations and see how
many of them have six 1's in a row.  You don't actually have to list the
combinations, just look at the number of bits. There are fifteen possible
places for six 1-bits to go, and each of them has 14 bits left over,
which we don't care about. So that's 15 * 2^14 out of 2^20 bit patterns
that meet the criteria. Rounding off a bit (literally), that's 2^18 /
2^20 or about 1/4. The two answers are slightly different because the
questions are slightly different: the first calculation gives a lower
probability because it's the probability of getting exactly six heads,
while the combinatorial solution gives the probability of getting at
least six heads.

-- Bruce


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