Every year on April 1, Anytown Tigers and Someville Lions play a soccer game. It

Question

Every year on April 1, Anytown Tigers and Someville Lions play a soccer game. It is always a high-scoring game; the number of goals scored follows a Poisson process with the average rate of one goal per 5 minutes. (A soccer game consists of two halves, 45 minutes each.) What is the probability that the fourth goal is scored during the last 10 minutes of the first half? (Enter at least 3 significant digits) What is the probability that the fifth goal is scored during the last 15 minutes of the first half? Find the probability that more than 17 goals are scored during the game. (Use Normal approximation.)

Explanation / Answer

Possion Distribution
PMF of P.D is = f ( k ) = e- x / x!
Where   
= parameter of the distribution.
x = is the number of independent trials

b.
during the first 30 minuets it scored only 4 goals
mean rate of goals in 30 minuets = 30/5 = 6

P( X = 4 ) = e ^-6 * 6^4 / 4! = 0.1339
P( atleast 1 goal in last 15 min) = P( X > = 1 ) = 1 - P (X < 1) = 0.9502
& 5th goals has pushed in last 15 minuete of first half is = P( 4 Goals in 30 Min) * P( atleast 1 goal in last 15 min) = 0.1339 * 0.9502 = 0.12723

c.
The rate of goal is 1 per 5 minute

mean rate of goal for 90 minute period is = 90/5 = 18
P( X < = 17) = P(X=17) + P(X=16) + P(X=15) + P(X=14) + P(X=13) ...P(X=0)
= e^-18 * 18 ^ 17 / 17! + e^-18 * 1 ^ 16 / 16! + e^-18 * ^ 15 / 15! + e^-18 * ^ 14 / 14! + e^-18 * ^ 13 / 13! +....P( X = 0 ) +e ^-18 * 18^0 / 0! = 0.468
P( X > 17) = 1 -P ( X <= 17) = 1 - 0.4686 = 0.5314

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