Probability

[reply2spg]

In a room filled with 7 people, 4 people have exactly 1 sibling in the room and 3 people have exactly 2 siblings in the room. If two individuals are selected from the room at random, what is the probability that those two individuals are NOT siblings?

5/21
3/7
4/7
5/7
16/21

Please explain

SPOILER: OA E - Manhattan GMAT test question.

[Brent Hanneson]

Originally Posted by reply2spg

In a room filled with 7 people, 4 people have exactly 1 sibling in the room and 3 people have exactly 2 siblings in the room. If two individuals are selected from the room at random, what is the probability that those two individuals are NOT siblings?

5/21
3/7
4/7
5/7
16/21

Please explain

SPOILER: OA E - Manhattan GMAT test question.

First, the question essentially tells us the 7 people look like this ABC DE FG, where the same color means the people are siblings.
If we select 2 people at random, what is probability that the two are NOT siblings?

Let's determine the probability that they ARE siblings. We'll, do this using counting techniques.
a) 2 siblings can be selected from the 3 blue people in 3 ways (3C2=3)
b) 2 siblings can be selected from the 2 red people in 1 way (2C2=1)
c) 2 siblings can be selected from the 2 green people in 1 way (2C2=1)

So there are 5 (3+1+1) ways to select 2 people who are siblings.
In total, there are 21 (7C2=21) ways to select 2 people from 7 people.

So, the probability that the two selected people ARE siblings = 5/21
Therefore, the probability that the two selected people are NOT siblings = 1 - 5/21 = 16/21 (E)

[Attack]

Consider AB, CD, EFG as the set of siblings as per the conditions.
consider the first person selected is A, (probablilty = 1/7)
hence probablity that B is selected next after A is selected = (1/7)*(1/6)
Similarly if B is selected first, the probablity that A is selected next = (1/7)*(1/6)
Since these are independent events, the probability that out of 7 the 2 selected are A and B = (1/7)*(1/6) + (1/7)*(1/6)= 1/21

in the same way for CD also it is 1/21

If the first person selected is E, then probablity that the next is either F or G is (1/7)* (2/6)
total probablity for EFG is 3*(1/7)* (2/6)= 3/21

Hence the probablity that the 2 peoople picked up are siblings is 1/21 + 1/21 + 3/21 = 5/21
Probablity that they are not siblings = 16/21

[carotene]

noob question here. what does this mean: (3C2=3)

I've searched the forum for a definition w/o any luck. thanks!


Update:

Nevermind, i found the answer: 3 choose 2 = 3
X! / (x-y)! / y!

[Attack]

Hi carotene, may be i can help.

3C2 is the representation of combination of 3 things taken 2 at a time.

nCm = n! / ((m!)(n-m)!)

en.wikipedia.org/wiki/Combination