Random Numbers with Constraints
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.
Generate two random numbers from 1 to 9, inclusive. Call them n1 and n2. What is the probability that | n1 - n2 | > 5?
Since you are making two choices of 9 numbers each, there are 81 total possible combinations (9 x 9).
The only combinations that have a difference greater than 5 are:
9 and 1, 2, or 3
8 and 1 or 2
7 and 1
This makes six combinations, but you also have to take them in the reverse order (i.e. 1 and 9 is a different combination than 9 and 1). This makes a total of 12 possibilities, making the probability 12 in 81.
Sep 14, 2003
|Huh? Oh wait... || means absolute value. It's been so long that I forgot about that.|
Sep 15, 2003
|I like that. Simple but makes you think. Do me another one!|
Dec 02, 2003
|I liked that one, too. You can simplify 12/81, I think. |
Aug 01, 2004
|Yes, you can. To 4/27.|
Nov 07, 2005
|I only came up with half as many. 9-1 is greater than 5, but 1-9 is not.|
Mar 27, 2006
|i think that isn't called combination. if 1-9 is the same as 9-1 then it is combination. but if 1-9 is not the same as 9-1 then it called permutation.|
Dec 17, 2008
|The answer is 9/64 = 0.140625.|
The answer 12/81 (0.148148148...) WOULD be correct IF the problem had stated two random integers instead of two random numbers.
With two random numbers, it's only possible for the difference to be greater than 5 if the lower number is less than 9-5 = 4 and the higher number is greater than 1+5=6. Each of these has a (4-1)/(9-1) = 3/8 probability. Either the first or second number can be in the low range, giving a (3/^2 * 2 = 18/64 probability. Of these cases, if n is the low number then the probability varies with from 1 to 0 as n moves from 1 to 4. This gives an average probability of 1/2, so the probability that |n1 - n2| > 5 is (18/64)*(1/2) = 9/64.
Dec 17, 2008
|Arrg! Of course instead of the smiley face above it should read [3/8]^2*2 = 18/64|
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