perplexus.info :: Probability : Box of Chocolates

Box of Chocolates (Posted on 2012-03-02)

A certain brand of candy is sold in boxes containing a specific number, lower than 50, of chocolates--a mix of creams and nut clusters, always the same number of each in each box.

If you choose one candy at a time at random without considering its type (cream or nut cluster), you are four times as likely to have one nut cluster left after eating the last cream as you are to have two nut clusters left after the last cream.

But also, if you were to flip a coin before eating each piece, to decide whether to eat a cream or a nut cluster, and then specifically choose one of those, it would be true that you'd be four times as likely to have X nut clusters left after eating the last cream as you'd be to have X+1 nut clusters left after the last cream, where of course X is a number smaller than the number of nut clusters in the box.

How many of each type are in each such box?

Submitted by Charlie

Rating: 4.0000 (1 votes)

Solution: (Hide)

The numbers that fit for each of the choice methods are:
Random selection among all the candies:

total creams nut clusters 6 4 2 10 7 3 14 10 4 18 13 5 22 16 6 26 19 7 30 22 8 34 25 9 38 28 10 42 31 11 46 34 12 Coin flip for type to choose: total creams nut clusters X 18 8 10 9 18 15 3 1 19 8 11 10 19 15 4 2 20 8 12 11 20 15 5 3 21 8 13 12 21 15 6 4 22 8 14 13 22 15 7 5 23 8 15 14 23 15 8 6 24 8 16 15 24 15 9 7 25 8 17 16 25 15 10 8 26 8 18 17 26 15 11 9 26 22 4 1 27 8 19 18 27 15 12 10 27 22 5 2 28 8 20 19 28 15 13 11 28 22 6 3 29 8 21 20 29 15 14 12 29 22 7 4 30 8 22 21 30 15 15 13 30 22 8 5 31 8 23 22 31 15 16 14 31 22 9 6 32 8 24 23 32 15 17 15 32 22 10 7 33 8 25 24 33 15 18 16 33 22 11 8 34 8 26 25 34 15 19 17 34 22 12 9 34 29 5 1

The only set of numbers of the two types that is on both lists is the 30-candy box, with 22 creams and 8 nut clusters, and in the second scenario, the numbers of nut clusters left are 5 and 6 respectively. In the first scenario the probability is 0.2022989 for one nut cluster to be left and 0.0505747 for two, it being most likely (0.733...) that no nut clusters will be left when eating the last cream. In the second scenario, the probabilities are much smaller that one or two nut clusters will be left: 0.0001206398 and 0.0000301600 respectively.

Adapted from Enigma No. 1676, "Pick'n'mix", by Susan Denham, New Scientist, 10 December 2011, page 32.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Answer	K Sengupta	2022-05-13 11:55:02
Life is like ...	Steve Herman	2012-03-03 11:50:30
an answer	Dej Mar	2012-03-03 09:09:19

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