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Two different numbers are selected simultaneously and at random from the set {1, 2, 3, 4, 5, 6, 7. What is the probability that the positive difference between the two numbers is 2 or greater? Express your answer as a common fraction.

 Jul 15, 2020
 #1
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Two different numbers are selected simultaneously and at random from the set {1, 2, 3, 4, 5, 6, 7. What is the probability that the positive difference between the two numbers is 2 or greater? Express your answer as a common fraction.  

 

I can grasp just from thinking that any two numbers will work if they're not consecutive.  That is, 1,2 won't work, nor 2,3, nor 3,4, etc.  

I don't know how to figure probabilities, so I have to make a grid and count them.  In the grid below, the colored pairs won't work. 

 

1,2   1,3   1,4   1,5   1,6   1,7

2,1   2,3   2,4   2,5   2,6   2,7

3,1   3,2   3,4   3,5   3,6   3,7

4,1   4,2   4,3   4,5   4,6   4,7

5,1   5,2   5,3   5,4   5,6   5,7

6,1   6,2   6,3   6,4   6,5   6,7

 

There are 36 pairs, and 11 of them won't work.  That means 25 of them will work.  So your probability is 25/36.

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 Jul 15, 2020
 #2
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I don't know much about Sets or Subsets, but it seems to me you may have overcounted the numbers in the Set, especially when he/she says "Two different numbers are selected simultaneously and at random from the set", which sounds to me that a pairing of 2 numbers, say {2,5} is the same as{5,2}. In other words, you have:

7 nCr 2 = 21 distinct ways  of choosing your numbers as oulined below:

 

{1, 2} | {1, 3} | {1, 4} | {1, 5} | {1, 6} | {1, 7} | {2, 3} | {2, 4} | {2, 5} | {2, 6} | {2, 7} | {3, 4} | {3, 5} | {3, 6} | {3, 7} | {4, 5} | {4, 6} | {4, 7} | {5, 6} | {5, 7} | {6, 7} (total: 21). And the differences between them would be:

 

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6 >> for a total of 21. 15 of them have a difference of 2 and >.

Then, according tto this scenario, the probability would be: 15 / 21 = 5/7

Note: Somebody more versed in Set Theory should take a look at this question. Thanks.

 Jul 15, 2020
 #3
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Good work.  You have a persuasive argument about the "selected simultaneously" condition, so I won't even address that, much less dispute it.  I said I didn't know how to figure probabilities.  But compare the answers we got... yours was 5/7 and mine was 25/36.  If mine were 25/35 instead, we would have the same answer since 25/35 reduces to 5/7.  I don't know what this signifies, if anything.   

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Guest Jul 15, 2020
 #4
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Actually, 5/7 is right. If I read the question right, you can tell that only 2 consecutive numbers have a difference smaller than 2. We know that there are 6 consecutive pairs of numbers in the set, and 7C2 = 21 total pairs. So the probability that the pair isn't consecutive is 1 - 6/21 which is 15/21 or 5/7. I think you misread the question and put the negative answers too, when the question says "positive difference" .

 

 

 

 - Pie

 Jul 23, 2020

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