Probability-siblings

lilu wrote: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

IMO E


There are 2 scenarios here :

1. Select 2 persons from group of 4 ,

Prob of selecting 2 person from group of 4 = 2C1 * 2C1 = 4

2. Select 1 from group of 4 and 1 from group of 3

Prob of selecting 1 person from group of 4 and 1 from group of 3 = 4C1*3C1 = 12

Total prob = 7C2 = 21

Therefore Ans is, Case 1 + Case 2 /21

16/21

can anyone give a better explanation for thsi please...

lilu wrote: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

probability of not selecting siblings= 1- (probability of selecting siblings);
we have 7 people 4 of which have exactly 1 sibling and 3 have exactly 2 siblings..!!!
let 7 people be p1,p2,p3,p4,p5,p6,p7; now out of these 7 peoples 4 people having exactly 1 sibling can be (p1&p2), (p3&p4); and people having exaclty two siblings can be (p5,p6,p7).

lets call them group 1(p1&2), group2(p3&p4) and group 3(p5,p6,p7)
out of 7 people any two people can be selected in 7C2 ways=21;

probability that two persons selected belongs to group1 =2C2=1;
probability that two persons selected belongs to group 2=2C2=1;
probability that two persons selected belongs to group 3=3C2=3;

hence the probability that selected persons will be sibling= 1+1+3/21=5/21;

required probability= 1-5/21=16/21; hence E

Consider the following sibling pairs S1, S2, and S3
S1 (A,B) S2(C,D) S3(E,F,G)

Now the probability of selecting A is 1C1. If A is selected then B cannot be selected and every other person can be selected. Thus the pairs can be AB, AC, AD, AE, AF, AG where the order of the selection doesn’t matter. So the probability of this selection is 1C1 * 5C1 / 7C2.
In the same way, B can also be paired with 5 other persons but not A.
BC, BD, BE, BF, BG. So the selection probability is 1C1*5C1/7C2.
Now consider the selection of C. CE, CF, CG i.e. 1C1*3C1/7C2 (since CA and CB have already been counted).
Same with D. DE, DF, DG i.e. 1C1*3C1/7C2 (Since DA, DB have already been counted).
However when we select E we cannot select the persons F and G, since they are kins. Also we cannot select any other person from the 4 people since that pair has already been counted above.
Same goes for F and G. Thus the total probability of selection is
5C1/7C2 + 5C1/7C2 + 3C1/7C2 + 3C1/7C2 = 16C1/7C2 = 16/21