Quote:
Originally Posted by twistedmosaic
So as I sit hear eating a starburst 2-pack (of which there is an entire drawer in the office), I started wondering about the probabilities they represented (clearly a busy day at work).
For people not familiar, there are four flavors, pink red orange and yellow, and each pack has two candies. The packs do have a left and right orientation, so in theory you could have 16 different candy combinations, if you consider 'right pink/left yellow' different from 'right yellow/left pink'.
Easy question: What is the probability that if you gathered 16 packs from a large population, you would have one of each combination?
Harder question: What is the probability that you would get one pack of each possible combination, if (like a sane person) you count 'right pink/left yellow' and 'right yellow/left pink' as being the same?
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Are you assuming the starbursts colors and orientation have a completely equal distribution? Without that assumption your first 2 questions are impossible to answer.
Quote:
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Harder question abstracted to dice: What is the probability of rolling 11 times and getting each number, 2 through 12, exactly once?
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This one is pretty easy is you generalize it for order.
Spoiler:
First you need to calculate the odds of rolling this in order.
To do this you multiply the odds of rolling 2-12 individually and then multiply by the number of orders you can roll it.
odds for 7 6/36 =0.167
odds for 6 and 8 5/36 =0.139
odds for 5 and 9 4/36 =0.111
odds for 4 and 10 3/36 =0.083
odds for 3 and 11 2/36 =0.056
odds for 2 and 12 1/36 =0.028
Multiply them together and you get:
(6*5^2*4^2*3^2*2^2)/36^11=86400/131621703842267136
this is the probability of rolling in a specific order. Now you have one of these for each possible way to order the rolls.
To compute the number of orders you will need to use a partial factorial.
The first roll has 11 possibilities (2-12). After each roll the number of possibilities drops by one. Thus you have a partial factorial. 11*10*9*8*7*6*5*4*3*2=39916800
Multiplying the odds for one order above yields a probability of: 86400*39916800/131621703842267136=2.62024531 × 10-5
or 1 in 38,164.3656.
It has been a few years since I have done any probability & stats how did I do?