Question

In the Zambia vs. Mali under 20 soccer match, the Zambian team scored, an average of one goal every 15 minutes interval of time.

        What is the probability that, the Zambia team will score:

Less than 3 goals per 15 minute interval? At least 3 goals per 15 minute interval?

Exactly 5 goals in a half hour interval? Between 3 goals and 6 goals in one hours?

Answer 1

Poisson distribution:

$P(x=k)=\frac{\lambda^ke^{-\lambda}}{k!}$

we have:

mean: $\lambda=1$ goal in 15 minutes

then:

$P(x<3)=P(x=0)+P(x=1)+P(x=2)$

$P(x=0)=e^{-1}=0.3679$

$P(x=1)=e^{-1}=0.3679$

$P(x=2)=\frac{e^{-1}}{2}=0.1839$

$P(x<3)=0.3679+0.3679+0.1839=0.9198$

$P(x\ge3)=1-P(x<3)=1-0.9198=0.0802$

for half hour interval:

$\lambda_1=2\lambda=2$

$P(x=5)=\frac{2^5e^{-2}}{5!}=0.0361$

for one hour:

$\lambda_1=4\lambda=4$

$P(3<x<6)=P(x=4)+P(x=5)$

$P(x=)=\frac{4^4e^{-4}}{4!}=0.1954$

$P(x=5)=\frac{4^5e^{-4}}{5!}=0.1563$

$P(3<x<6)=0.1954+0.1563=0.3518$