Here is the problem: There is a fair coin. We flip the coin 10 times, and we look for the probability to have equal appearences of [H]eads and [T]ails.
Here is my solution: The probability will be P=|A|/|W| where
A={all sequences of 5H and 5T} W={all possible outcomes from flipping a coin 10 times}
=> Since every time we flip a coin we have two possible outcomes and we repeat that 10 times, then |W|=2^10 .
=> Now the outcomes that we want, are those that will have 5H and 5T (since we look after for probability of equals appearences of H and T, and since the experiment is repeated 10 times the equality means 5H and 5T). Let's imagine:
H H H H H T T T T T [outcome]
= = = = = = = = = =
1 2 3 4 5 6 7 8 9 10 [experiment]
so i look for: what are the permutations of the above ordering? I can find in how many ways i can assign the numbers to the 5H (and the rest empty slots will be assigned to Ts)=>
10 9 8 7 6
= = = = = --> 30240
H H H H H
and the rest empty slots are filled with the Ts.
So P = 30240/1024 > 1 !!
What is wrong here?
p.s: Intuitively, since the coin is fair, and there is a symmetry (5H, 5T) the P has to be 1/2.