Question

PROBLEM 2 Two teams A and B play a soccer match. The number of goals scored by Team A is modeled by a Poisson process Ni(t) with rate l1 = 0.02 goals per minute, and the number of goals scored by Team B is modeled by a Poisson process N2(t) with rate 12 = 0.03 goals per minute. The two processes are assumed to be independent. Let N(t) be the total number of goals in the game up to and including time t. The game lasts for 90 minutes. (a) Find the probability that no goals is scored, i.e., the game ends with a 0-0 draw. (b) Find the probability that at least two goals are scored in the game. (c) Find the probability of the final score being team A: 1, team B : 2 (d) Find the probability that they draw.

Answer 1

N(t) = N1(t) + N2(t) ~ Poisson(lambda), lambda = 0.03+0.02 = 0.05/min

a) P(N = 0) = (lambda*90)^0*exp(-lambda*90)/0!

= 0.0111

b) P(N >= 2) = 1 - P(N < 2)

= 1 - (P(N = 0) + P(N = 1))

= 1 - 0.0111 - (lambda*90)^1*exp(-lambda*90)/1!

= 1 - 0.0111 - 0.0499

= 0.939

C) P(N1 = 1, N2 = 2) = P(N1 = 1)*P(N2 = 2)

= ((lambda1*90)^1*exp(-lambda1*90)/1!) * ((lambda2*90)^2*exp(lambda2*90)/2!)

= 0.5425

D) P(draw) = P(N1 = N2 = k(say)), k = 0, 1, 2, ....to infinity

2 P. (2. = N₂ =0) + P (N = N2 = 2) + .... so P (N, = 0) P (N2 = 0) + PIN, = 1) P (N2=1) + ... .. - (2, ₂) expet | () (9) + (-() ) ) ) ), (3) P (A,A1) 01 s expl-Cd, +22)90] (Ald2 90236 + 1:0

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