Assume the number of commuters using the QR code payment service at ea

Ann Tice

Ann Tice

Answered question

2021-11-18

Assume the number of commuters using the QR code payment service at each MTR station follows a Poisson probability distribution. Based on a recent statistics, on average, 3 commuters use the QR code payment service in an hour.
(i) What is the probability there are exactly 10 commuters who use the QR code payment service in an hour?
(ii) What is the probability that there are less than 4 commuters who use the QR code payment service in 3 hours?
(iii) If five MTR stations are randomly selected, what is the probability that at least two of the five MTR stations will have less than 4 commuters who use the QR code payment service in 3 hours?

Answer & Explanation

pseudoenergy34

pseudoenergy34

Beginner2021-11-19Added 22 answers

Step 1
Given information:
μ=3
i) x=10
By applying poisons distribution:
P(x,μ)=(eμ)(μx)x!
Step 2
Use the excel formula, "=POISSON.DIST(10,3,FALSE)" for determining the probability:
P(x=10)=0.0008
The probability there are exactly 10 commuters who use the QR code payment service in an hour is 0.0008.
ii) Average number of commuters in 3 hours =33
=9
x=4
By applying poisons distribution:
P(x,μ)=(eμ)(μx)x!
P(x<4)=P(x<3)
Use the excel formula, "=POISSON.DIST(10,3,FALSE)" for determining the probability:
P(x<4)=0.0212
The probability that there are less than 4 commuters who use the QR code payment service in 3 hours is 0.0212.
Step 3
iii) P(x<4)=0.0212
n=5
x=2
By applying binomial distribution:
P(x,n)=nCxpx(1p)(nx)
P(x>2)=1P(x<1)
Use the excel formula, "=BINOM.DIST(1,5,0.0212,TRUE)" to determine the probability.
P(x>2)=10.9957
P(x>2)=0.0043
The probability that at least two of the five MTR stations will have less than 4 commuters who use the QR code payment service in 3 hours is 0.0043.

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