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.

Guest Jul 15, 2020

#1**0 **

*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**.

_{.}

Guest Jul 15, 2020

#2**0 **

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.

Guest Jul 15, 2020

#3**0 **

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.

_{.}

Guest Jul 15, 2020

#4**0 **

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

AppIePi Jul 23, 2020