lagomez wrote:

You have people 1-2-3-4-5-6-7

4 have exactly 1 sibling which can mean:

1-2 are siblings

3-4 are siblings

3 have exactly 2 siblings which can mean:

5-6-7 are siblings

Let's start with the group of 4:

The probability of picking 1 is (1/7)

The probability of not getting a sibling pair is (5/6) because the only other sibling is 2

Therefore, if 1 is selected first the probability of not getting a sibling pair is 5/42

Multiply that by 4 because the probability is the same whether you start with 1-2-3-4 so you get 20/42 for the first 4 people

Now let's go to the group of 3:

The probability of picking 5 is (1/7)

The probability of not getting a sibling pair is (4/6) which the non-sibling pair is 1-2-3-4

Therefore the probability is 4/42. Multiply that probability by 3, which represent 5-6-7 so the probability is 12/42

Now you have two probabilities: 12/42 and 20/42

add both and you get 32/42 or 16/21

It's easier to use the complement. Not sure why people are trying so hard here to use the choose formula when it makes the question 1000X harder

4/7 chance to pick a person with 1 sibling * 1/6 chance you get a sibling or 4/42

3/7 change to pick a 2 person sibling * 2/6 or 6/42

Total of 10/42 of getting a sibling.

We get to use the simple or rule not the generalized since the things are mutually exclusive. And we aren't drawing twice just once. So either a 4 person is picked first or 3 person.

So that simplifies to 5/21 then 1 minus this is 16/21 for the odds that they are not siblings. 5/21 they are.