There are 20 people in your applicant pool, including 5 pairs of identical twins.

If you hire 5 people randomly, what are the chances you will hire at least 1 pair of identical twins? (Needless to say, this could cause trouble ;))

Submitted

Answer

The probability to hire 5 people with at least 1 pair of identical twins is 25.28%

5 people from the 20 people can be hired in 20C5 = 15504 ways.

Now, divide 20 people into two groups of 10 people each :

G1 - with all twins

G2 - with all people other than twins

Let's find out all possible ways to hire 5 people without a single pair of indentical twins.

People from G1 People from G2 No of ways to hire G1 without a single pair of indentical twins No of ways to hire G2 Total ways

0 5 10C0 10C5 252

1 4 10C1 10C4 2100

2 3 10C2 * 8/9 10C3 4800

3 2 10C3 * 8/9 * 6/8 10C2 3600

4 1 10C4 * 8/9 * 6/8 * 4/7 10C1 800

5 0 10C5 * 8/9 * 6/8 * 4/7 * 2/6 10C0 32

Total 11584

Thus, total possible ways to hire 5 people without a single pair of indentical twins = 11584 ways

So, total possible ways to hire 5 people with at least a single pair of indentical twins = 15504 - 11584 = 3920 ways

Hence, the probability to hire 5 people with at least a single pair of indentical twins

= 3920/15504

= 245/969

= 0.2528

= 25.28%

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