6 balls in a bag, draw 3, chance of getting the only white ball?

[ruijoel]

I can't make sense out of this one.

I have a bag with 5 black balls and 1 white ball. I draw 3 balls. What's the probability of drawing the white ball?

Intuitively, I'd say it's 50%. But I calculated that 1/6 + 1/5 + 1/4 = 62%, so I should have better than a 50/50 chance of getting the white ball.

What's the correct answer? If it's not 50%, is there a name for the fallacy of thinking that it's 50% ?

 

[HallsofIvy]

The probability that the white ball will be in the three balls drawn is 1 minus the probability it is not.
If it is not, then we must have gotten all black balls.

Initially, there are 6 balls, 5 of them black. The probability the first ball drawn is black is 5/6. If that happens, there are 5 balls, 4 of them black. The probability the second ball drawn is black is 4/5. Finally, given that the first two balls drawn are black, there are 4 balls left, 3 of them black. The probability the last ball drawn is black is 3/4. The probability all 3 balls are black is (5/6)(4/5)(3/4)= 3/6= 1/2. The probability that the 3 balls are two black, one white, is 1- 1/2= 1/2, just as you guessed.