Question

The number of people arriving at an emergency room follows a Poisson distribution with a rate of 7 people per hour.

a. What is the probability that exactly 5 patients will arrive during the next hour? (3pts)

b. What is the probability that at least 5 patients will arrive during the next hour? (5pts)

c. How many people do you expect to arrive in the next two hours? (4pts)

d. One in four patients who come to the emergency room in a hospital. Calculate the probability that during the next 2 hours exactly 20 people will arrive and less than 7 will be hospitalized (8pts)

Answer 1

The number of people arriving at an emergency room follows a Poisson distribution with a rate of 7 people per hour.

Let 'X' be variable for the number of people arriving

( = 7) per hour

=

a. What is the probability that exactly 5 patients will arrive during the next hour? (3pts)

P(X = 5) =

Ans: 0.12772

b. What is the probability that at least 5 patients will arrive during the next hour? (5pts)

At least 5 patients means 5 or more

P( X >= 5) = 1 - P( X < 5)

=1 - [ P(X = 0) + P( X = 1) + P( X = 2) + P( X = 3) + P( X = 4)]

We sub different 'x' in each and find the answers

= 1 - (0.0091 + 0.0064 + 0.0223 + 0.0521 + 0.0912)

Ans: 0.82701

c. How many people do you expect to arrive in the next two hours? (4pts)

The possion is rate per period. So we can get rate for different periods. The current rate is 7 per hour. So for two hours it will be 7 * 2 = 14

E(X in 2 hours) = 14

d. One in four patients who come to the emergency room in a hospital. Calculate the probability that during the next 2 hours exactly 20 people will arrive and less than 7 will be hospitalized (8pts)

This requires two steps

1st to find the probability where there are 20 people arriving

2nd to find the probability that less than 7 will be hospitalized given 20 people arrived in 2 hours

( = 14) per 2 hours

=

P(X = 20) = 0.0286

Now we need to know that out of 20 , less than 7 will be hospitalized. The chances of one person to be hospitalized is 1 in 4 which = 1/4 = 0.25

This is like a binomial experiment where you either get hospitalized or don't.

(n = 20 ,p = 0.25 )

P( X =x) =

                = 20Cx * 0.25^x * 0.75^(20-x)

P(X < 7) = P( X = 0) + .......P( X = 6)

X	P(X=x)
0	0.0032
1	0.0211
2	0.0669
3	0.1339
4	0.1897
5	0.2023
6	0.1686
Total	0.7858

P(X < 7) = 0.7858

P (less than 7 will hospitlized | given 20 arrive in next 2 hours) = P( X = 20) * P( less than 7 hospitalized)

= 0.0286 * 0.7858

Ans: 0.02247