What is more probable – getting at least one six in four throws of a die or getting at least one double six in 24 throws of a pair of dice?
It is a paradox because common sense (again!) tells you that both are equally probable. The probability of getting a six for a single die is (1/6), and that for two sixes from a pair of dice is (1/6)x(1/6). So by extrapolation, what happens in four throws for a single may become six times more (24) for double dice.
Well, the answer is wrong. Here is the calculation.
One dice
One dice
A) The probability of getting a six in one roll is (1/6).
B) The probability of getting no six in a roll is, therefore, (5/6).
C) The probability of getting no sixes in four rolls is (5/6)4 = 0.48.
D) The chance of getting at least one six in four throws is 1 – 0.48 = 0.52.
A pair
Following the steps above
A) The probability of getting a double-six in a pair of rolls is (1/6)x(1/6) = 1/36.
B) The probability of getting no double-six in a pair of rolls is 35/36.
C) The probability of getting no double-six in 24 rolls of a pair of rolls is (35/36)24 = 0.51.
D) The chance of getting at least one double-sixes in 24 rolls of a pair of dice is 1 – 0.51 = 0.49.
In summary
Getting one six in four rolls is more probable than getting one double-six in twenty-four.
