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

Step by step

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) = μ^x × 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