This is not as mathematically interesting, but I always feel a good old sledgehammer can show that for small numbers anyone can do maths....


We know that the series must finish THH and any series of ten tosses will finish this way 1/8 of the time.


Now we only have to look at seven tosses, out comes the hammer.

There are 128 possible tosses:


0 heads: 1 combination

1 heads: 7 combinations

2 heads: 21 combinations

3 heads: 35 combinations

4 heads: 35 combinations

5 heads: 21 combinations

6 heads: 7 combinations

7 heads: 1 combination


Picture the individual tosses as a series of boxes labelled:

1 2 3 4 5 6 7


for 0 and 1 heads, we know there are never two heads together.

for two heads, there are 6 ways of arranging so that two are together. (12,23,34,45,56,67)

for three heads there are 30 ways, but five are duplicated as triples, so 25 ways. (1,2*5,2,3*5 etc)

for four heads there is only one way not to have them together (1,3,5,7)

for five, six and seven, obviously none


so there are 0+0+6+25+34+21+7+1 ways = 94 combinations from 128 where there are heads together.

Therefore only 34/128 of the combinations have no heads together for seven tosses.

and we said at the start that this only works once in eight, so we get the figure 34/1024 as already shown.


I hope this inspires some hope in those with a passing interest in probability and only a light foundation in maths (I have massive interest in probability and a light foundation in maths [img]/images/smile.gif[/img] )