>I know this to be true but there is some part of me that wonders whether I would bet against the person who says that out of a hundred tosses at least one of them will be heads and at least on of them will be tails. Seems that there must be some other level of probability that could be determined. The odds for tossing 100 straight heads or 100 straight tails could not be the same as tossing 50 heads and 50 tails. Could it? Seems improbable... <g>
>

It's not. The issue is when you make the call. The odds for 100 straight heads before you start is 1 in 2^100. The odds for 50 straight heads before you start is 1 in 2^50.

Here's the math behind it. Assume throughout that you're dealing with a fair coin (one that isn't weighted in some way to change the results).

When you flip a coin once, there are two possible, equally likely, outcomes, heads (H) or tails (T). So the chance of each is 1 in 2.

When you flip a coin twice, there are 4 possible outcomes, as follows:

HH
HT
TH
TT

If you don't care about the order, that reduces to three outcomes, not equally likely:

2 heads - 1 in 4
2 tails - 1 in 4
one of each - 2 in 4

When you flip a coin 3 times, there are 8 possible outcomes:

HHH
HHT
HTH
HTT
THH
THT
TTH
TTT

Again, when you ignore order, it reduces to 4 possibilies, with their probabilities:

3 heads - 1 in 8
3 tails - 1 in 8
2 heads, 1 tails - 3 in 8
2 tails, 1 heads - 3 in 8

You can continue the process, and what you see is that no matter how many coin flips are involved, there's only one way to get all heads. The total number of arrangements is 2^(number of flips).

What makes all these feel wrong intuitively is that people have been talking about two different cases. In one case, none of the results are known; in the other case, all but the last flip has already happened.

So, when you're starting from scratching, there 1 chance in 2^100 of 100 straight heads. However, after 99 flips that all come up heads, there's 1 chance in 2 of 100 straight heads. That's because only two arrangements out of 2^100 are still possible, the one with 100 heads and the one with 99 heads followed by one tails.

Tamar