You don’t get exactly 1⁄3 of a win with no variance in either case. You get exactly 1 win, 1⁄3 of the time, and no win 2⁄3 of the time.
As an example when betting on green, suppose there’s a 1⁄3 chance of 30 blue and 30 green balls, 1⁄3 chance of 60 green, 1⁄3 chance of 60 blue. And there’s always 30 red balls.
There is a 1⁄3 of 1⁄3 chance that there are 30 green balls and you pick one. There is a 2⁄3 of 1⁄3 chance that there are 60 green balls and you pick one. There is no chance that there are no green balls and you still pick one. Therre is no other way to get a green ball. The total chance of picking a green ball is therefore 1⁄3, that is, 1⁄3 of 1⁄3 plus 2⁄3 of 1⁄3. That means that 1⁄3 of the time you win and 2⁄3 of the time you lose, just as in the case of betting on the red ball.
A distribution of 1 one third of the time and 0 two thirds of the time has some computable variance. Whatever it is, that’s the variance in your number of wins when you bet on green, and it’s also the variance in you number of wins when you bet red.
You don’t get exactly 1⁄3 of a win with no variance in either case. You get exactly 1 win, 1⁄3 of the time, and no win 2⁄3 of the time.
As an example when betting on green, suppose there’s a 1⁄3 chance of 30 blue and 30 green balls, 1⁄3 chance of 60 green, 1⁄3 chance of 60 blue. And there’s always 30 red balls.
There is a 1⁄3 of 1⁄3 chance that there are 30 green balls and you pick one. There is a 2⁄3 of 1⁄3 chance that there are 60 green balls and you pick one. There is no chance that there are no green balls and you still pick one. Therre is no other way to get a green ball. The total chance of picking a green ball is therefore 1⁄3, that is, 1⁄3 of 1⁄3 plus 2⁄3 of 1⁄3. That means that 1⁄3 of the time you win and 2⁄3 of the time you lose, just as in the case of betting on the red ball.
A distribution of 1 one third of the time and 0 two thirds of the time has some computable variance. Whatever it is, that’s the variance in your number of wins when you bet on green, and it’s also the variance in you number of wins when you bet red.