at least one six with 6 dice
at least two sixes with 12 dice
at least three sixes with 18 dice
at least four sixes with 24 dice
Answer: 6 dice.
The probability for one six with 6 dice is .665, 2 from 12 = .619, 3 from 18 = .597 etc. To explain:
For one dice, the probability is simply the probability of not getting a six on any dice, subtracted from 1, so:
1 - ((5/6)^6) = 0.665 (3 dp)
For two sixes, we subtract the probability of no sixes on 12 dice and the probability of exactly one six on 12 dice from 1. The probability of one six is (5/6)^11 (for 11 non-sixes) times (1/6) (for one six) times 12 (for the 12 different arrangements of 1 six and 11 non-sixes - we have to count all these because (5/6)^11*(1/6) is the probability that dice 1 is six and the other 11 are not six, but dice 2, or dice 3, or any of the others could be the one that is the six), so:
1 - (5/6)^12 - (5/6)^11*(1/6)*12 = 0.619 (3 dp)
Similarly for three sixes:
1 - (5/6)^18 - (5/6)^17*(1/6)*18 - (5/6)^16*(1/6)^2*18*17/2 = 0.597 (3 dp).
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