Question

10. What is the probability (give in %) that in a group of 3 people A. No two people have the birthday in the same month? B. At least two people have the birthday in the same month?

Answer 1

A.

For  a group of 3 people, number of possible ways to have birthday months = 12 * 12 * 12 = 12³ =  1728 (Each person can have birthday in any of 12 months)

Number of possible ways to have birthday months so that no people share the birthday months = ¹²P₃ = 12! / (12 - 3)!

= 12! / 9! = 12 * 11 * 10 = 1320

Probability that in a group of 3 people, no two people have the birthday in the same month = Number of possible ways to have birthday months so that no people share the birthday months / number of possible ways to have birthday months in 3 people

= 1320 / 1728

= 0.7639

= 76.39%

B.

Probability that at least two people have the birthday in the same month = 1 - Probability that no people have the birthday in the same month

= 1 - 0.7639 = 0.2361

= 23.61%