Arrivals during the evening at a hospital emergency room are

Arrivals during the evening at a hospital emergency room are assumed to follow a Poisson distribution. The rate on Saturday evening is 2.5 per hour and on Sunday evening is 0.5 per hour.

a.) Find the probability of 6 or more arrivals on Saturday during the period 8pm to midnight.

b.) Find the probability of 2 or fewer arrivals on Sunday during the period 9pm to midnight, given that there is at least 1 arrival.

Solution

for poisson distribution

P(X=k) = (r*t)^k*e^(-r*t)/k!

r = rate

t = time interval

so,

a)

here , t = 4 hours

r = 2.5

P(X>=6) = 1-P(X<6) = 1-{P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)}

= 1-{(2.5*4)^0*e^(-2.5*4)/0! + (2.5*4)^1*e^(-2.5*4)/1! + (2.5*4)^2*e^(-2.5*4)/2! + (2.5*4)^3*e^(-2.5*4)/3! + (2.5*4)^4*e^(-2.5*4)/4! + (2.5*4)^5*e^(-2.5*4)/5! }

= 0.9329

b)

here,

r = 0.5

t = 3 hours

so,

now,

P(X>=1 AND X<=2) = P(1<=X<=2) = P(X=1) + P(X=2)

= (0.5*3)^1*e^(-0.5*3)/1! + (0.5*3)^2*e^(-0.5*3)/2!

= 0.5857

also,

P(X>=1) = 1 - P(X=0) = 1-(0.5*3)^0*e^(-0.5*3)/0! = 0.77687

so, by conditional probability definition,

P(X<=2 | X>=1) = P(X>=1 AND X<=2) / P(X>=1) = 0.5857/0.77687 = 0.7539 = 0.754