> 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?"
>
> I believe that the number of tosses (20) doesn't matter.

Sure it does:

The number of possible result sequences from 20 tosses is 2exp20 (duuh!)

Of those, 120 have 6 consecutive heads somewhere in the sequence, as
follows:

 12345678901234567890
 HHHHHH
  HHHHHH
   HHHHHH
    HHHHHH
     HHHHHH
      HHHHHH
       HHHHHH
        HHHHHH
         HHHHHH
          HHHHHH
           HHHHHH
            HHHHHH
             HHHHHH
              HHHHHH
               HHHHHH
 15 sequences of exactly 6 consecutive heads
 HHHHHHH
  HHHHHHH
   HHHHHHH
    HHHHHHH
     HHHHHHH
      HHHHHHH
       HHHHHHH
        HHHHHHH
         HHHHHHH
          HHHHHHH
           HHHHHHH
            HHHHHHH
             HHHHHHH
              HHHHHHH
 14 sequences of exactly 7 consecutive heads
 13 sequences of exactly 8 consecutive heads
 12     "      "    "    9      "        "
 11     "      "    "   10      "        "

You get the idea. The formula is
 sequences = ((t - r + 1)exp2 + (t - r + 1)) / 2
where t = # of tosses and r = consecutive run length. This is the standard
  n + (n-1) + (n-2) +...+ 1 = 1/2 * (nexp2 + n)

With 120 sequences that contain 6 consecutive heads somewhere, the
probability is
  120/2exp20 = 0.00011444091796875

If there were only 10 tosses, the probability would be
  15/2exp10 = 0.0146484375

And, of course, the most obvious reason why the number of tosses matters: 6
tosses. The probability is 1/2exp6 = 0.015625, more than two orders of
magnitude larger than with 20 tosses.

Generally:
 with t = number of tosses
      r = desired run length
 let n = t - r + 1
 p = (nexp2 + n)/2exp(t + 1)

So, p is very definitely dependent on t (and also on r).


Steve Dirickson   WestWin Consulting
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