Is the Probability of Getting an Even Number of Heads 1/2 After 491 Coin Tosses?

[damightytom]

Homework Statement

A fair coin is tossed 491 times. The total number of heads or tails is then even or uneven.

Is the probability that the head will result in an even result equal to 1/2
Motivate your answer with a strict mathematical proof.

Homework Equations

I am having some trouble grasping the solution of this.
Im struggleing on the finish line and I could use some help.

First thing I would like to get help with is to understand how I can calculate the this sum.
I might be able to continue on proving the end goal myself but at this moment I'm pretty stuck.

The Attempt at a Solution

So, since there's a 1/2 chance for each toss.
You can use the binomial distribution equation to find the sum of all the probably outcomes that heads will be even.

p = (0.5)^n \sum {k=0,2,4,6,8,...,490} C(n,k) * 0.5^n * 0.5^(n-k)

 

[HallsofIvy]

Since the probability of head or tails is equal on each flip, the probability that the total number of heads is even, the total number of heads is odd, the total number of tails is even, and the total number of tails is odd are all the same- call that "p". Since further, if the total number of heads is even, then the total number of tails must be odd, we must have p+ p= 2p= 1.