Steply

The Poisson distribution

University Statistics · 23 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Name the numbers
  2. Write the formula
  3. Put the numbers in
  4. Work it out
  5. Check

The steps for Poisson mean 2, P(X = 3). Every solution names its steps, so you always know what comes next.

Worked examples

Poisson mean 2, P(X = 3) Easy

The question

Question: X is Poisson with mean 2: find the probability that X is exactly 3.

Poisson, μ = 2: P(X = 3)?

Poisson counts how many times something happens in a fixed time, when we know the average.

Poisson: counts in a time or a space, with a known average

1 Name the numbers

μ, the average, is 2.

μ = 2

x, the count we want, is 3.

x = 3

2 Write the formula

The Poisson formula: μ to the power x, times e to the power minus μ, over x factorial.

P(X = x) = μx × e−μx!

e is a number, about 2.718.

e to the power minus 2 is about 0.1353.

x factorial multiplies down to 1, and 0 factorial is 1.

e ≈ 2.718, e−2 ≈ 0.1353
x! = x × ... × 2 × 1, 0! = 1

Careful!

0 factorial is 1, not 0.

Careful: 0! = 1 (not 0)

3 Put the numbers in

2 to the power 3 is 8.

3 factorial is 3 times 2 times 1: 6.

23 = 8, 3! = 3 × 2 × 1 = 6

4 Work it out

8 times 0.1353, over 6, is about 0.1804.

P(X = 3) = 23 × e−23! = 8 × 0.13536 ≈ 0.1804

5 Check

Let's check another way: step up from P of 0.

P of 0 is e to the power minus 2: about 0.1353.

P(0) = e−2 ≈ 0.1353

Each next one is the last one times 2, divided by the new x.

P of 1 is about 0.2707.

P(1) = P(0) × 2 ÷ 1 ≈ 0.2707

P of 2 is about 0.2707.

P(2) = P(1) × 2 ÷ 2 ≈ 0.2707

P of 3 is about 0.1804.

The same!

So it is right!

P(3) = P(2) × 2 ÷ 3 ≈ 0.1804

Let's look back.

First: name the numbers.

Then: write the formula.

Then: put the numbers in.

Then: work it out.

Last: check.

Answer
P(X = 3) = 23 × e−23! = 8 × 0.13536 ≈ 0.1804

▶ Watch it solved

A shop gets an average of 4 calls an hour, probability of exactly 2… Medium

The question

Question: a shop gets an average of 4 calls an hour: what is the probability of exactly 2 calls in an hour?

average 4 an hour: exactly 2 in an hour?

Poisson counts how many times something happens in a fixed time, when we know the average.

Poisson: counts in a time or a space, with a known average

1 Name the numbers

μ, the average, is 4.

μ = 4

x, the count we want, is 2.

x = 2

2 Write the formula

The Poisson formula: μ to the power x, times e to the power minus μ, over x factorial.

P(X = x) = μx × e−μx!

e is a number, about 2.718.

e to the power minus 4 is about 0.0183.

x factorial multiplies down to 1, and 0 factorial is 1.

e ≈ 2.718, e−4 ≈ 0.0183
x! = x × ... × 2 × 1, 0! = 1

Careful!

0 factorial is 1, not 0.

Careful: 0! = 1 (not 0)

3 Put the numbers in

4 to the power 2 is 16.

2 factorial is 2 times 1: 2.

42 = 16, 2! = 2 × 1 = 2

4 Work it out

16 times 0.0183, over 2, is about 0.1465.

P(X = 2) = 42 × e−42! = 16 × 0.01832 ≈ 0.1465

5 Check

Let's check another way: step up from P of 0.

P of 0 is e to the power minus 4: about 0.0183.

P(0) = e−4 ≈ 0.0183

Each next one is the last one times 4, divided by the new x.

P of 1 is about 0.0733.

P(1) = P(0) × 4 ÷ 1 ≈ 0.0733

P of 2 is about 0.1465.

The same!

So it is right!

P(2) = P(1) × 4 ÷ 2 ≈ 0.1465

Let's look back.

First: name the numbers.

Then: write the formula.

Then: put the numbers in.

Then: work it out.

Last: check.

Answer
P(X = 2) = 42 × e−42! = 16 × 0.01832 ≈ 0.1465

▶ Watch it solved

A shop gets an average of 6 calls an hour, probability of exactly 2… Harder

The question

Question: a shop gets an average of 6 calls an hour: what is the probability of exactly 2 calls in 30 minutes?

average 6 an hour: exactly 2 in 30 minutes?

Poisson counts how many times something happens in a fixed time, when we know the average.

Poisson: counts in a time or a space, with a known average

1 Name the numbers

The average is for one hour.

30 minutes is 0.5 hours.

So the average for our time is 6 times 0.5: 3.

μ = 6 × 0.5 = 3

Careful!

The average must be for the same time as the question: 3, not 6.

Careful: not μ = 6 ✗

x, the count we want, is 2.

x = 2

2 Write the formula

The Poisson formula: μ to the power x, times e to the power minus μ, over x factorial.

P(X = x) = μx × e−μx!

e is a number, about 2.718.

e to the power minus 3 is about 0.0498.

x factorial multiplies down to 1, and 0 factorial is 1.

e ≈ 2.718, e−3 ≈ 0.0498
x! = x × ... × 2 × 1, 0! = 1

3 Put the numbers in

3 to the power 2 is 9.

2 factorial is 2 times 1: 2.

32 = 9, 2! = 2 × 1 = 2

4 Work it out

9 times 0.0498, over 2, is about 0.2240.

P(X = 2) = 32 × e−32! = 9 × 0.04982 ≈ 0.2240

5 Check

Let's check another way: step up from P of 0.

P of 0 is e to the power minus 3: about 0.0498.

P(0) = e−3 ≈ 0.0498

Each next one is the last one times 3, divided by the new x.

P of 1 is about 0.1494.

P(1) = P(0) × 3 ÷ 1 ≈ 0.1494

P of 2 is about 0.2240.

The same!

So it is right!

P(2) = P(1) × 3 ÷ 2 ≈ 0.2240

Let's look back.

First: name the numbers.

Then: write the formula.

Then: put the numbers in.

Then: work it out.

Last: check.

Answer
P(X = 2) = 32 × e−32! = 9 × 0.04982 ≈ 0.2240

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. Poisson mean 2, P(X = 3)
    Answer
    P(X = 3) = 23 × e−23! = 8 × 0.13536 ≈ 0.1804
  2. Poisson mean 3, P(X = 2)
    Answer
    P(X = 2) = 32 × e−32! = 9 × 0.04982 ≈ 0.2240
  3. Poisson mean 1, P(X = 0)
    Answer
    P(X = 0) = 10 × e−10! = 1 × 0.36791 ≈ 0.3679
  4. Poisson mean 4, P(X = 2)
    Answer
    P(X = 2) = 42 × e−42! = 16 × 0.01832 ≈ 0.1465
  5. Poisson mean 2, P(X = 0)
    Answer
    P(X = 0) = 20 × e−20! = 1 × 0.13531 ≈ 0.1353
  6. Poisson mean 5, P(X = 3)
    Answer
    P(X = 3) = 53 × e−53! = 125 × 0.00676 ≈ 0.1404
  7. P(X = 1), Poisson mean 3
    Answer
    P(X = 1) = 31 × e−31! = 3 × 0.04981 ≈ 0.1494
  8. Poisson λ = 2, P(X = 1)
    Answer
    P(X = 1) = 21 × e−21! = 2 × 0.13531 ≈ 0.2707

Medium

  1. a shop gets an average of 4 calls an hour, probability of exactly 2 calls in an hour
    Answer
    P(X = 2) = 42 × e−42! = 16 × 0.01832 ≈ 0.1465
  2. a road has an average of 3 accidents a week, probability of exactly 1 accident in a week
    Answer
    P(X = 1) = 31 × e−31! = 3 × 0.04981 ≈ 0.1494
  3. a bakery sells an average of 5 cakes an hour, probability of exactly 3 cakes in an hour
    Answer
    P(X = 3) = 53 × e−53! = 125 × 0.00676 ≈ 0.1404
  4. Poisson mean 3, P(X <= 2)
    Answer
    P(X ≤ 2) = 0.0498 + 0.1494 + 0.2240 ≈ 0.4232
  5. Poisson mean 2, P(X <= 1)
    Answer
    P(X ≤ 1) = 0.1353 + 0.2707 ≈ 0.4060
  6. Poisson mean 2, P(X >= 1)
    Answer
    P(X ≥ 1) = 1 − 0.1353 ≈ 0.8647
  7. Poisson mean 4, P(X >= 1)
    Answer
    P(X ≥ 1) = 1 − 0.0183 ≈ 0.9817
  8. Poisson mean 1.5, P(X = 2)
    Answer
    P(X = 2) = 1.52 × e−1.52! = 2.25 × 0.22312 ≈ 0.2510

Harder

  1. a shop gets an average of 6 calls an hour, probability of exactly 2 calls in 30 minutes
    Answer
    P(X = 2) = 32 × e−32! = 9 × 0.04982 ≈ 0.2240
  2. a website gets an average of 2 orders a minute, probability of exactly 5 orders in 2 minutes
    Answer
    P(X = 5) = 45 × e−45! = 1024 × 0.0183120 ≈ 0.1563
  3. a help desk gets an average of 12 emails a day, probability of at least 1 email in 2 hours
    Answer
    P(X ≥ 1) = 1 − 0.3679 ≈ 0.6321
  4. Poisson mean 3, P(X >= 2)
    Answer
    P(X ≥ 2) = 1 − (0.0498 + 0.1494) ≈ 0.8009
  5. Poisson mean 2.5, P(X <= 2)
    Answer
    P(X ≤ 2) = 0.0821 + 0.2052 + 0.2565 ≈ 0.5438
  6. Poisson mean 1, P(X <= 3)
    Answer
    P(X ≤ 3) = 0.3679 + 0.3679 + 0.1839 + 0.0613 ≈ 0.9810
  7. Poisson mean 4, P(X >= 3)
    Answer
    P(X ≥ 3) = 1 − (0.0183 + 0.0733 + 0.1465) ≈ 0.7619