There is a bag containing one thousand balls - yes, it will be a big bag! Each of the balls is numbered like lottery balls, from one to one thousand.
If two balls are drawn at random, is the probability of the sum of the numbers on the two balls being even . . .
Greater than the sum being odd?
Less than the sum being odd?
The same as the sum being odd?
For the sum of the two balls to be even either both balls are odd numbered or both balls are even numbered. For the sum of the two balls to be odd one must be odd numbered and one must be even numbered.
Whether an odd numbered ball or an even numbered ball is drawn first there will be 499 balls with the same odd/evenness and 500 balls with different odd/evenness.
Thus the chances of an even sum are 499/999 and the chances of an odd sum are 500/999, so the probability of and even sum is less than an odd sum.