"Getting back the stake" is a given when you win and isn't counted toward winnings. If you win yougain$1. If you lose, youlose$1. Imagine doing this 1000 times. Because "the probability of getting an odd number is equal to that of even number" you would expect to get around 500 odd numbers and 500 even numbers. That is, you would expect to win $500 and lose $500, netting $0. This is a "fair game".

If you have given this correctly, the expected value is neither $1.00 nor $0.50, it is 0.so what is expected value of game?

the answer was something $1 (it should be correct)

but I don't quite understand... my argument is ....

while if I lose, then I will lose the stake, then it will be -$1

if I win, then i will get back $2 in total, so it should be $2

1/2 ($2) + 1/2 (-$1) = $1/2 = $0.5

I really don't quite understand how $1 comes even though my teacher had explained to me, she was very unclear