Question

The number of arriving customers to a big supermarket is following a Poisson distribution with a rate of 4 customers per a minute.

What is the probability that no customer will arrive in a given minute?

What is the probability that exactly 3 customers will arrive in a given minute?

What is the probability that at least seven customer will arrive in a given minute?

What is the probability that at most one customer will arrive in 40 seconds?

What is the average number of arriving customers in a given one hour?

Answer 1

Solution:

Let X be a random variable which represents the number of customers arriving to a big supermarket.

Given that X follows poisson distribution with mean rate of 4 customers per minute.

According to Poisson process, Probability of occurrence of exactly x events in time t is given by,

Where, is mean rate of occurrence of the event.

We have, customers/minute

1) We have to find P(X = 0) in t = 1 minute

λ = 4 customers/minute and t = 1 minute

Hence, λt = 4

Using Poisson probability law we get,

The probability that no customer will arrive in any given minute is 0.0183.

2) We have to find P(X = 3) in t = 1 minute

λ = 4 customers/minute and t = 1 minute

Hence, λt = 4

Using Poisson probability law we get,

The probability that exactly three customer will arrive in any given minute is 0.1954.

3) have to find P(X = at least 7) in t = 1 minute

λ = 4 customers/minute and t = 1 minute

Hence, λt = 4

P(X = at least 7) = P(X ≥ 7)

P(X ≥ 7) = 1 - P(X < 7)

P(X ≥ 7) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)]

Using Poisson probability law we get,

The probability that at least 7 customer will arrive in any given minute is 0.1107.

4) We have to find P(X = at most one) in t = 40 seconds

λ = 4 customers/minute and t = 40 seconds = 40/60 = 2/3 minutes

Hence, λt = 4 × (2/3) = 8/3

P(X = at most one) = P(X ≤ 1)

P(X ≤ 1) = P(X = 0) + P(X = 1)

Using Poisson probability law we get,

The probability that at most customer will arrive in 40 seconds is i0.2548.

5) Average number of arriving customer in an hour is given by,

Average = λt

λ = 4 customers/minute, t = 1 hour = 60 minutes

Hence, average = 4×60 = 240

Average number of arriving customer in an hour is 240.

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