Is the probability of an even number of tails in 491 coin tosses exactly 50%?

[kaitoli]

Hi everyone

There is a question which I find very hard to solve and it goes like this..

A symmetric coin with heads on one side and tails on the other side is tossed 491 times after one another. The total amount of times you get tails is either even or odd. Is the probability that you get an even amount of tails exactly 50%? And the question requires a strong mathematical evidence, e.g. a formula.

 

[mathman]

Mathematical proofs don't always require formulas. In this case, the basic argument is symmetry (assuming a fair coin). Therefore the probability of n tails and m heads is the same as the probability of m tails and n heads, where n and m are arbitrary with n+m=491.