A website gets an average of 2 orders a minute, probability of… Step by Step

Step by step

The question

Question: a website gets an average of 2 orders a minute: what is the probability of exactly 5 orders in 2 minutes?

average 2 a minute: exactly 5 in 2 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 minute.

2 minutes is 2 minutes.

So the average for our time is 2 times 2: 4.

μ = 2 × 2 = 4

Careful!

The average must be for the same time as the question: 4, not 2.

Careful: not μ = 2 ✗

x, the count we want, is 5.

x = 5

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) = μ^x × 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

3 Put the numbers in

4 to the power 5 is 1024.

5 factorial is 5 times 4 times 3 times 2 times 1: 120.

4⁵ = 1024, 5! = 5 × 4 × 3 × 2 × 1 = 120

4 Work it out

1024 times 0.0183, over 120, is about 0.1563.

P(X = 5) = 4⁵ × e⁻⁴5! = 1024 × 0.0183120 ≈ 0.1563

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.

P(2) = P(1) × 4 ÷ 2 ≈ 0.1465

P of 3 is about 0.1954.

P(3) = P(2) × 4 ÷ 3 ≈ 0.1954

P of 4 is about 0.1954.

P(4) = P(3) × 4 ÷ 4 ≈ 0.1954

P of 5 is about 0.1563.

The same!

So it is right!

P(5) = P(4) × 4 ÷ 5 ≈ 0.1563

Let's look back.

First: name the numbers.

Then: write the formula.

Then: put the numbers in.

Then: work it out.

Last: check.

Answer