536 views
8 pairs of shoes are in a closet. 8 shoes are selected at random. The probability that there will be at least one pair and at most 3 pairs amongst selected shoes are?

1 - ((8C4+28)/16C8) = 0.9746

@papesh is this correct??? I'm not sure...

good approach

1st case:we have taken 1 pair.then from 8 pairs,we can choose it in 8 ways.now we have to take another 6 shoes from remaining 7 pairs, which will not be pair of anyone.this will be 7C6 * 26 =7*64=448(choosing 6 pairs from 7. after that taking any one of them from each pair).so total ways=(8*448)=3584

2nd case:we have taken 2 pairs.then from 8 pairs,we can choose it in 8C2=28 ways.now we have to take another 4 shoes from remaining 6 pairs, which will not be pair of anyone.this will be 6C4 * 24 =15*16=240(choosing 4 pairs from 6. after that taking any one of them from each pair).so total ways=(28*240)=6720

3rd case:we have taken 3 pairs.then from 8 pairs,we can choose it in 8C3=56 ways.now we have to take another 2 shoes from remaining 5 pairs, which will not be pair of anyone.this will be 5C2 * 22=10*4=40(choosing 2 pairs from 5. after that taking any one of them from each pair).so total ways=(56*40)=2240

so required probability=(3584+6720+2240)/16C8=6272/6435=0.9746

S = 16C8;
A = (2C2x14C6)+(4C4x12C4)+(6C6x10C2) = 3543;
P = A/S = 0.275.
Explanation: Firstly think the probability has to come very less to 1,beacause der r 16 shoes and the pairs r being randomly chosen,so definitely 0.947 is not the answer!LOL!
Now just see der r 16 shoes(8 pairs) ::::
The meaning of 8 shoes randomly selected is,u can choose either 4 pairs or in random way
Now atleast 1 and atmost 3 pairs means, among 16 u can choose 2 shoes(which has to be a pair)
so 2C2,and remaining 6 shoes r randomly selected from 14,so the event for atleast one is::::2C2x14C6
I think this is simple question! and d rest of all is the same way!!!

@shailesh11

I think there is a flaw in this logic because in this term 14C6 u are selecting any 6 shoes from 14 shoes. Now in this selection we may have 1more pair or 2 more pairs or 3 more pairs. Also you are counting them thrice because of terms 12C4 and 10C2.

Also i don't know why u find the answer 0.9746 funny. In selecting 8 shoes from 16 shoes chances of getting no pair and chances of getting all 4 pairs is very low.16C8 is 12870.In case of no pair, Possiblities of getting no pair are 2^8=256 because from each pair we can select only one shoes i.e. either left or right but not both, 8 such pairs are there.Hence, 2^8. Fraction 256/12870 is 0.019 which is very low. Similarly, to get all 4 pairs, we have to select 4 pairs from 8 given shoes pairs. So 8C4=70 which is 70/12870.

Subtracting the sum of these probabilities from 1 gives 0.9746

Hence the answer of atleast one and atmost three is closer to 1.
Sorry!,Ive solved it in 5 mins,didnt notice that in 14 there might be pairs can come out,but after asserting this logic,i might get the answer.Thanks for reply.
can anyone suggest me that how would i can improve my probabality >?????????????????????

http://gatecse.in/combinatory/

Specially IITM link for counting is very helpful in when to use what formula and imagining situation in each problem in terms of balls and bins

For conditional probability

https://youtu.be/JGeTcRfKgBo

This MIT video will clear all ur concepts

Hope it helps

1
199 views