Question

Cargo ships arrive at a loading dock at a rate of 2 per day. The dock has the capability of handling 3 arrivals per day. How many days per month (assume 30 days in a month) would you expect the dock being unable to handle all arriving ships? (Hint: first find the probability that more than 3 ships arrive and then use that probability to find the expected number of days in a month too many ships arrive.)

Answer 1

4 days

Explanation:

We need to use the probability functions of each of the intervals to know the Probability number and then use it in the expected value.

P(x>3 cargo ships) = 1-P(x<=3)

P(x>3) = 1-P(x<=3)

P(x>3) = 1-[P(x=0)+P(x=1)+P(x=2)+P(x=3)]

`P(x>3) = 1 - [(e^(-2)2^0)/(01)+(e^(-2)2^1)/(11)+(e^(-2)2^2)/(21)+(e^(-3)2^3)/(31)]`

P(x>3) = 1- [0.1353(1+2+2+1.33)]

P(x>3) = 1-0.856

P(x>3) = 0.1431

Als n=30, expected number is

E(x) =30*P(x>3)

E(x) = 30*0.1431

E(x) = 4.293

I expect 4.293 or 4 days per month the block being unable to hanlde all arriving ships