Probability Question

chrisjim5 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?

Can you please provide an answer to this?

Let's the 7 people be ABCDEFG.

4 people have exactly 1 sibling:

Let A and B are siblings and C and D are siblings.

So,
A has 1 sibling B
B has 1 sibling A
C has 1 sibling D
D has 1 sibling C

3 people have exactly 2 siblings:

Let E, F and G are all siblings of each other.

So,
E has 2 siblings (F and G).
F has 2 siblings (E and G).
G has 2 siblings (E and F).

Total number of sibling pairs = 5 [AB, CD, EF, EG, FG]

Total number of pairs that can be formed from 7 people C(7,2) = 7!/5!*2! = 21

P(sibling pair) = 5/21

P(not sibling pair) = 1 - 5/21 = 16/21.

Can someone please tell me where am i going wrong!!!

7 people,

4 have 1 siblings i.e. (a1, a2) and (b1, b2)
3 have 2 siblings i.e. (c1,c2,c3)



total ways of selecting two people out of 7 = 7c2

favorable ways:

(ways of selecting one from Ci *(ways of selecting one from left 4people) + (ways of selecting one from Ai *(ways of selecting one from left 5 people) + (ways of selecting one from Bi *(ways of selecting one from left 5 people)

= 3c1*4c1 + 2c1*5c1 + 2c1*5c1

P(no siblings) = fav ways/total ways = (3c1*4c1 + 2c1*5c1 + 2c1*5c1) / 7c2

= 2( 16/21)

waltz2salsa wrote:Can someone please tell me where am i going wrong!!!

7 people,

4 have 1 siblings i.e. (a1, a2) and (b1, b2)
3 have 2 siblings i.e. (c1,c2,c3)



total ways of selecting two people out of 7 = 7c2

favorable ways:

(ways of selecting one from Ci *(ways of selecting one from left 4people) + (ways of selecting one from Ai *(ways of selecting one from left 5 people) + (ways of selecting one from Bi *(ways of selecting one from left 5 people)

= 3c1*4c1 + 2c1*5c1 + 2c1*5c1

P(no siblings) = fav ways/total ways = (3c1*4c1 + 2c1*5c1 + 2c1*5c1) / 7c2

= 2( 16/21)

You're overcounting the number of non-sibling pairs:

Number of ways to combine 1 choice from (c1,c2,c3) and 1 choice from (a1,a2,b1,b2) = 3*4 = 12.

We've now counted all the non-sibling pairs that include (c1, c2, c3). We need to count only the non-sibling pairs that don't include (c1,c2,c3).
Number of ways to combine 1 choice from (a1,a2) and 1 choice from (b1,b2) = 2*2 = 4.

Total non-sibling pairs = 12+4 = 16.

Total possible pairs from 7 people = 7C2 = 21.

P(non-sibling pair) = 16/21.