Probability

P( 2 equal birthday in group of 15 ) ≈ 22.3%

The probability that 2 persons and only 2 persons in a random group of n persons can be calculated using the following expression*:

P(2 equal bd in group of n persons) = nC2(1-1/1461)(4/1461)Π2n-1[1-4(i-1)/1461] + nC2(1/1461)2Π2n-1[1-(4i-3)/1461]

with n=15, 30C2 = 105,

P(2 eq bd in group of 15) = 105(1-1/1461)(4/1461)[1-4/1461]

[1-8/1461][1-12/1461]∙∙∙[1-52/1461]

P(...) = 0.223263549... ≈ 22.3%

*You find this expression discussed in question "What is the probability that in a room of 8 people 2 have the same birthday ?". A simpler form of this expression that neglects February 29 of the leap day that gives a good

approximation is:

P(2 eq bd in group of n persons) = nC2(1/365)Π1n-1[1-(i-1)/365]

for n=15, this expression gives P(...) = 0.214918798... ≈ 21.5%

The mathematics of the collection, organization, and interpretation of numerical data, especially the analysis of population characteristics by inference from sampling.

Read more: statistics

the odds theoretically are almost infinity to 1 as they could be any color. -------------------------------------------------------------------------------------------
2nd opinion

Let's say we have 3 boxes with 4 balls each.

Box A has 4 white balls.

Box B has 3 white balls.

Box C has 2 white balls.

The probability of drawing 2 W balls from;

Box A, P(2W│A)=

(4/4)∙(3/3)=

1

Box B, P(2W│B)=

(3/4)∙(2/3)=

1/2

Box C, P(2W│C)=

(2/4)∙(1/3)=

1/6

Say the probability of picking any of the 3 boxes is the same, we have;

P(A)=

1/3

P(B)=

1/3

P(C)=

1/3

Question is, given the event of drawing 2 W balls from a box taken blindly

from the 3 choices, what is the probability that the balls came from box A,

P(A│2W).

Recurring to Bayes Theorem:

P(A│2W)=

[P(A)P(2W│A)]/[P(A)P(2W│A)+P(B)P(2W│B)+P(C)P(2W│C)]=

[(1/3)(1)]/[(1/3)(1)+(1/3)(1/2)+(1/3)(1/6)]=

6/10=

0.60=

60%

P(A│2W)=

0.60=

60%

Read more:Solution_to_a_bag_contains_4_balls_Two_balls_are_drawn_at_random_and_are_found_to_be_white_What_is_the_probability_that_all_balls_are_white