A help desk gets an average of 12 emails a day, probability of at… Step by Step

Step by step

The question

Question: a help desk gets an average of 12 emails a day: what is the probability of at least 1 emails in 2 hours?

average 12 a day: at least 1 in 2 hours?

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 day.

2 hours is 0.0833 days.

So the average for our time is 12 times 0.0833: 1.

μ = 12 × 0.0833 = 1

Careful!

The average must be for the same time as the question: 1, not 12.

Careful: not μ = 12 ✗

At least 1: x can be 1, 2, 3, and so on, with no end.

x = 1, 2, 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) = μ^x × e^−μx!

e is a number, about 2.718.

e to the power minus 1 is about 0.3679.

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

e ≈ 2.718, e⁻¹ ≈ 0.3679

x! = x × ... × 2 × 1, 0! = 1

3 Use the rest

x can go on with no end, so we cannot add them all.

All the probabilities add up to 1: so we take away the ones below 1.

P(X ≥ 1) = 1 − P(X < 1)

For 0: 1 to the power 0 is 1, and 0 factorial is 1.

1 times 0.3679, over 1, is about 0.3679.

P(X = 0) = 1⁰ × e⁻¹0! = 1 × 0.36791 ≈ 0.3679

1 minus 0.3679 is about 0.6321.

P(X ≥ 1) = 1 − 0.3679 ≈ 0.6321

4 Check

Let's check the key number another way: e to the power 1 is about 2.7183.

1 divided by it is about 0.3679: the same as e to the power minus 1.

So it is right!

e¹ ≈ 2.7183, 1 ÷ 2.7183 ≈ 0.3679

Let's look back.

First: name the numbers.

Then: write the formula.

Then: use the rest.

Last: check.

Answer