what is the probabilty of 10 people have different birthdays

Question

what is the probabilty of 10 people have differennt birthdays

Answer 1

If we use 365 as the number of days in a year and we have two people A and B. The probability of their birthdays not coinciding=364/365.

If we have 3 people A, B, C then if we use ≠ to mean they have different birthdays then we have the relationships and probabilities:

P(A≠B)=364/365, P(A≠C)=364/365, P(B≠C)=364/365. If we combine these probabilities we get (364/365)³.

For 4 people:

P(A≠B)=364/365, P(A≠C)=364/365, P(A≠D)=364/365, P(B≠C)=364/365, P(B≠D)=364/365, P(C≠D)=364/365. Combined probability (364/365)⁶.

For 5 people:

P(A≠B)=364/365, P(A≠B)=364/365, P(A≠B)=364/365, P(A≠B)=364/365, P(A≠B)=364/365, P(A≠B)=364/365, P(A≠B)=364/365, P(A≠B)=364/365, P(A≠B)=364/365, P(A≠B)=364/365. Combined probability=(364/365)¹⁰.

1; 1+2=3; 1+2+3=6; 1+2+3+4=10 for 2, 3, 4, 5 people. So the exponent is n(n-1)/2 where n is the number of people. When n=10, the combined probability is (364/365)⁴⁵=0.884 approx (88.4%).