Poisson Distribution Practice Problems (original) (raw)

Last Updated : 8 Jan, 2026

**Poisson distribution is a probability distribution that models the number of events occurring within a fixed interval of time or space, where these events happen with a known constant mean rate and independently of the time since the last event.

Poisson Distribution Formulas

The table below represents the important formulas of the Poisson distribution which is required to solve the Practice Questions on Poisson Distribution.

Poisson Distribution Practice Problems- Solved

**Question 1: If 2% of the products made in a company are defective. Find the probability that less than 1 item is defective in the sample of 100 items.

Here, n = 100, p = (2/100) = 0.02, q = 0.98, ƛ = np = 2

The formula for the required probability is given by:

P (X = x) = [ƛx × e-ƛ] / x!

P(X < 1) = P(X = 0)

P(X < 1) = [20 × e-2] / 0!

P(X < 1) = e-2

**Question 2: If the probability of choosing incorrect answer in an examination is 0.01, determine the chance that out of 500 students more than 1 will choose incorrect answer.

Here, n = 500, p = 0.01, ƛ = np = 5

The formula for the required probability is given by:

P (X = x) = [ƛx × e-ƛ] / x!

P(X > 1) = 1 - [P(X = 0) + P(X = 1)]

P(X = 0) = [50 × e-5] / 0! = e-5

P(X = 1) = [51 × e-5] / 1! = 5e-5

P(X = 0) + P(X = 1) = e-5 + 5e-5 = 6e-5

P(X > 1) = 1 - 6e-5

P(X > 1) = 0.960

**Question 3: A man can send an email on average 2 emails per hour. What is the probability that the man sends no email in a given hour?

Mean = 2

The formula for the required probability is given by:

P (X = x) = [ƛx × e-ƛ] / x!

P(X = 0) = [20 × e-2] / 0! = e-2

P(X = 0) = 0.135

**Question 4: Calculate the mean of the Poisson Distribution given that the number of trials is 20 and probability of success is 0.6.

The mean is the Poisson distribution is given by:

Mean = np

Here, n = 20 and p = 0.6

Mean = 20 × 0.6

Mean = 12

**Question 5: Find the mean of the Poisson distribution given that the variance of the Poisson distribution is 4.

We know that,

In Poisson distribution Mean = Variance = np

Mean = Variance

Mean = 4

**Question 6: Calculate the variance and standard deviation of the Poisson distribution given the number of trials as 50 and probability of failure as 0.3.

Formula for the variance and standard deviation in Poisson distribution is given by:

Variance = np and Standard Deviation = √(np)

Here, n = 50 and p = 1- q = 0.7

Variance = n × p

Variance = 50 × 0.7 = 35

Standard Deviation = √Variance

Standard Deviation = √35

Poisson Distribution Practice Problems - Unsolved

**Q1. If 5% of the product made in a company are defective. Find the probability that less than 3 item is defective in the sample of 200 items.

**Q2. If the probability of choosing incorrect answer in an examination is 0.001, determine the chance that out of 1000 students more than 3 will choose incorrect answer.

**Q3. A shop has an average of 10 customers per hour. What is the probability that exactly 12 customers arrive in an hour?

**Q4. Calculate the mean of the Poisson Distribution given that the number of trials is 100 and probability of success is 0.9.

**Q5. Find the mean of the Poisson distribution given that the variance of the Poisson distribution is 20.

**Q6. Calculate the variance and standard deviation of the Poisson distribution given the number of trials as 200 and probability of success as 0.8.