The mean number of oil tankers at a port city is 8 per day. The port has facilities to handle up to 12 oil tankers in day. What is the probability that too many tankers will arrive on a given day?

Question

Quinn Allen · Accepted Answer

Assuming (perhaps without justification) a Poisson distribution,
the probability of more than 12 tankers arriving on a given day is (approximately) #color(red)(6.38%)#

If the tankers arrive with a Poisson distribution (which seems likely but not certain), then
The probability of #color(blue)k# tankers arriving in a single day
given the average of #color(magenta)(lamda=8)# tankers per day
is given by the formula:
#color(white)("XXX")P(color(blue)k)=e^(-color(magenta)lamda) * (color(magenta)lamda^color(blue)k)/(color(blue)k!)=e^(-color(magenta)8) * (color(magenta)8^color(blue)k)/(color(blue)k!)#
and
the probability of #12# or fewer tankers arriving would be
#color(white)("XXX")P(<=12)=Sigma_(k=0)^12 e^(-8) * (8^k)/(k!)#

The probability of more than #12# tankers arriving would be
#color(white)("XXX")1-P(<=12)#

These equations can be evaluated with a calculator or (as below) by using a spreadsheet:

HIX Tutor · Answer

undefined To find the probability that too many tankers will arrive on a given day, you need to determine how many tankers arriving would be considered "too many."

In this case, the port can handle up to 12 tankers in a day, but the mean number of tankers is 8 per day. Let's assume that any number of tankers greater than 12 would be considered "too many."

To find the probability, follow these steps:

Step 1: Calculate the probability of having 0 to 12 tankers (inclusive) arriving on a given day.

Step 2: Subtract the probability of having 0 to 8 tankers (inclusive) arriving on a given day from the probability in Step 1.

Let's calculate the probability using these steps:

Step 1:
To find the probability of having 0 to 12 tankers (inclusive) arriving on a given day, you need to use the concept of a binomial distribution.

The formula to calculate the probability of having x successes in n trials (where n is the number of days and x is the number of tankers) is:
P(x) = (nCx) * p^x * (1-p)^(n-x)

In our case, the number of trials (n) is 1, and the probability of success (p) is 8/12.

Let's calculate the probabilities for having 0 to 12 tankers:

P(0) = (1C0) * (8/12)^0 * (4/12)^1 = (1) * (1) * (1/3) ≈ 1/3 ≈ 0.33
P(1) = (1C1) * (8/12)^1 * (4/12)^0 = (1) * (8/12) * (1) = 2/3 ≈ 0.67
P(2) = (1C2) * (8/12)^2 * (4/12)^0 = (0) * (64/144) * (1) = 0
P(3) = (1C3) * (8/12)^3 * (4/12)^0 = (0) * (512/1728) * (1) = 0
P(4) = (1C4) * (8/12)^4 * (4/12)^0 = (0) * (4096/6912) * (1) = 0
P(5) = (1C5) * (8/12)^5 * (4/12)^0 = (0) * (32768/46656) * (1) = 0
P(6) = (1C6) * (8/12)^6 * (4/12)^0 = (0) * (262144/2985984) * (1) = 0
P(7) = (1C7) * (8/12)^7 * (4/12)^0 = (0) * (2097152/23967488) * (1) = 0
P(8) = (1C8) * (8/12)^8 * (4/12)^0 = (0) * (16777216/191739552) * (1) = 0
P(9) = (1C9) * (8/12)^9 * (4/12)^0 = (0) * (134217728/1533916896) * (1) = 0
P(10) = (1C10) * (8/12)^10 * (4/12)^0 = (0) * (1073741824/12292177936) * (1) = 0
P(11) = (1C11) * (8/12)^11 * (4/12)^0 = (0) * (8589934592/98373786912) * (1) = 0
P(12) = (1C12) * (8/12)^12 * (4/12)^0 = (0) * (68719476736/789582010368) * (1) = 0

Step 2:
To find the probability of having too many tankers (more than 8), subtract the probabilities of having 0 to 8 tankers (inclusive) from the probability of having 0 to 12 tankers:

P(too many) = P(9) + P(10) + P(11) + P(12)
P(too many) = 0 + 0 + 0 + 0 = 0

Therefore, the probability that too many tankers will arrive on a given day is 0 or 0%.