Question

An office allows individuals to sign up for different group classes. They estimate that on average 1.5 individuals sign up per day and that this rate does not vary with day of the week and that there is independence among all individuals signing up and between days of the week.

What is the probability that exactly 2 individuals sign up on a random day?

What is the probability that 5 or more individuals sign up on a random day?

What is the probability that fewer than 3 individuals sign up on any given day?

What is the probability that no one signs up for 2 consecutive days?

Suppose it is known that on a certain day, fewer than 3 people signed up. What is the probability that no one signed up on this day?

Answer 1

1) probability that exactly 2 individuals sign up on a random day =e^-1.5*1.5²/2! =0.251021

2)

probability that 5 or more individuals sign up on a random day =P(X>=5)=1-P(X<=4)

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

=1-(e^-1.5*1.5⁰/0!+e^-1.5*1.5¹/1!+e^-1.5*1.5²/2!+e^-1.5*1.5³/3!+e^-1.5*1.5⁴/4!)

=1-0.981424=0.018576

3) probability that fewer than 3 individuals sign up on any given day =P(X<=2)

=P(X=0)+P(X=1)+P(X=2)

=e^-1.5*1.5⁰/0!+e^-1.5*1.5¹/1!+e^-1.5*1.5²/2!

=0.808847

4)expected number of signups in 2 days=2*1.5=3

probability that no one signs up for 2 consecutive days =e^-33⁰/0! =0.049787

5)

probability that no one signed up on this day given fewer than 3 people signed up

=P(X=0)/P(X<3)=P(X=0)/P(X<=2)=0.223130/0.808847=0.275862