Step by step
The question
Question: X is Poisson with mean 4: find the probability that X is at least 3.
Poisson counts how many times something happens in a fixed time, when we know the average.
1 Name the numbers
μ, the average, is 4.
At least 3: x can be 3, 4, 5, and so on, with no end.
2 Write the formula
The Poisson formula: μ to the power x, times e to the power minus μ, over x factorial.
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.
Careful!
0 factorial is 1, not 0.
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 3.
For 0: 4 to the power 0 is 1, and 0 factorial is 1.
1 times 0.0183, over 1, is about 0.0183.
For 1: 4 to the power 1 is 4, and 1 factorial is 1.
4 times 0.0183, over 1, is about 0.0733.
For 2: 4 to the power 2 is 16, and 2 factorial is 2.
16 times 0.0183, over 2, is about 0.1465.
1 minus the sum 0.0183 plus 0.0733 plus 0.1465 is about 0.7619.
4 Check
Let's check the key number another way: e to the power 4 is about 54.5982.
1 divided by it is about 0.0183: the same as e to the power minus 4.
And step up: each next P is the last one times 4, divided by the new x.
P of 1 is about 0.0733.
P of 2 is about 0.1465.
The same as above.
So it is right!
Let's look back.
First: name the numbers.
Then: write the formula.
Then: use the rest.
Last: check.