Question

Anja, Bengt and Carola play a game. What the game is about and how it works, we don't care. We only need to know that the highest score wins, that Anja's score is a Poisson-distributed stochastic variable with parameter 2, that Bengt's score is Poisson distributed with parameter 3, that Carola's score is Poisson distributed with parameter 4 and that the players' points are independent of each other . The score was 1, 2 and 3, but we don't know who got what. What is the conditional probability given to this information that it was Carola who won?

Answer 1

X = score by Anja

Y = score by bengt

Z = Carola

since Carol won hence score is 3 is hers

P((X = 1 , Y = 2 , Z = 3) or (X = 2 , Y = 1 ,Z = 3)) | (Z = 3))

as X, Y and Z are independent

required probability reduce to

P(X = 1 , Y = 2) + P(X = 2 , Y = 1)

X follow poisson with mean 2 , Y follow poisson with mean 3

P(X = 1)	0.270671
P(X = 2)	0.270671
P(Y = 1)	0.149361
P(Y = 2)	0.224042

formula in excel

P(X = 1)	=POISSON(1,2,0)
P(X = 2)	=POISSON(2,2,0)
P(Y = 1)	=POISSON(1,3,0)
P(Y = 2)	=POISSON(2,3,0)

P(X = 1 , Y = 2) + P(X = 2 , Y = 1)

= 0.270671*(0.149361 + 0.224042)

= 0.101069