Times Roll
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How many times to roll a die?
How many times would you expect to roll a fair die before all 6 sides appeared at least once?
This can be answered mathematically - you need to calculate the distribution of rolls to generate all six values, then calculate the mean number of rolls. It's a complex integral.
The probability of rolling all numbers in six rolls is very low. It's:
P(6) = (6/6) * 5/6 * 4/6 * 3/6 * 2/6 * 1/6 = 6! / 6^6 = 1/64.8 ~= 1.5%
I expect the mean will be somewhere around 13-15 rolls, but that's a guess - I'll work out the mathematical calculations later when I have time.
Ed: Ok, let's start by calculating the probability of rolling all six sides once in 7 rolls:
Number of allowed combinations: 6 (any number can be used twice)
Number of permutations: 6 * (7! / 2) {the numbers can appear in any order, but two can be swapped as duplicates.}
So that makes P(6 sides in 7 rolls) = (6*7!/2) / 6^7 = 7! / 2 * 6^6 ~= 0.054 = 5.4%
Now let's try 8 rolls:
Combinations: 6 + 6C2 {3 of one number, 2 of two numbers}
Permutations: 6*(8!/6) + 6C2 * 8! / (2*2)
P(6 sides in 8 rolls) = [8! + 6C2 * 8! / (2*2)] / 6^8 = 0.114 = 11.4%
Cumulative probability to this point is P(<= 8 rolls) = 1.5% + 5.4% + 11.4% = 18.3%
Let's keep going - this is converging quickly:
9 rolls:
Combinations: 6 + 6 * 5C2 + 6C3 {4 of one #, 3 of 1 & 2 of 1 #s, 2 of three #s}
Permutations: 6*(9!/24) + 6*5 (9! / 6*2) + 6C3 * 9! / (2*2*2)
P(6 sides in 9 rolls) ~= 362880 / 6^9 ~= 18.9%
Cumulative P(<= 9) ~= 35.2%
10 rolls:
Combinations: 6 + 6 * 5C2 + 6C3 {5 of one #, 4 of 1 & 2 of 1 #s, 3 of 2 #s, 3 of 1 & 2 of 2 #s, 2 of four #s}
Permutations: 6*(10!/120) + 6*5 (10! / 24*2) + 6*5 (10! / 6*6) + 6* 5C2 (10!/6*2*2) + 6C4 * 10! / (2*2*2*2)
P(6 sides in 10 rolls) ~= 3628800 / 6^10 ~= 29.7%
Cumulative P(<= 10) ~= 64.3%
So clearly, the median will occur when you reach the 10th roll. (This is surprising to me! It can be checked experimentally though.) The distribution of number of rolls to achieve this will have the median skewed to the left, so the mean will be higher than this median - I'm guessing the mean would be about 11-12. Experimentally, this matched what happened when I tried it.
Whats On My YouTube Today?
The Cars - Good Times Roll - Live 1978
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