How to find a Poisson probability

Find the probability of exactly two calls or at most two calls in a minute when the mean is three.

5 steps

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A call centre receives an average of three calls per minute. Using a Poisson model, what is the probability of exactly two calls in a minute? What about at most two? Let $X$ be the number of calls in one minute:

X \sim Poisson (3), P (X = 2), P (X \leq 2)

Here $x = 2$ is the number of calls to check, and $λ = 3$ is the mean number of calls in the same interval.

Find the probabilities of exactly two and at most two calls

Below each step is the screen you should see after pressing.

Next to each step is the screen you should see after pressing.

Step 1

Open the mode menu and tap Distribution.

Press:
Step 2

Tap the type name at the top of the screen to open the Select Type sheet.
Step 3

Tap Poisson PD. The parameter boxes appear with $x$ selected.
Step 4

Enter $2$ for $x$ and press the equals key to move to $λ$ . Enter $3$ for $λ$ . The answer updates to $0.224 0418077$ as you type; no further equals press is needed.

Press:
Step 5

Tap the type name again, then tap Poisson CD. The calculator keeps $x = 2$ and $λ = 3$ and shows $0.423 1900811$ .

Read and check the result

Poisson PD gives the probability of exactly two calls. The screen shows $0.224 0418077$ , or about $0.2240$ .

Poisson CD includes zero, one or two calls. The screen shows $0.423 1900811$ , or about $0.4232$ . You can check this by adding the three probabilities:

P (X \leq 2) = e^{- 3} (1 + 3 + \frac{3 ^{2}}{2 !}) = 8.5 e^{- 3} \approx 0.4232.

For "at least three calls" there is no key. Subtract the cumulative value from $1$ :

P (X \geq 3) = 1 - P (X \leq 2) = 1 - 0.4231900811 = 0.5768099189.

Below either answer, check that Mean and Variance both show $3$ : for a Poisson distribution, both equal $λ$ .

If your result is different

Argument ERROR beside a dashed box: $x$ must be a whole number, and $λ$ must be positive. For example, $x = 2.5$ or $λ = 0$ causes an error. Tap the affected box, press AC once to empty it, then enter the correct value. A second AC resets every box of the selected type to its defaults.
The answer is not $0.2240$ : after $x = 2$ and the equals key, the answer is live for whatever $λ$ is still in its box (the default $λ = 1$ gives $0.183 9397206$ ). Enter $3$ for $λ$ .
Old values or the wrong type appear: tap the type name and choose Poisson PD again, then enter both example values. Values are kept by name when you switch types, so choosing a type does not reset the boxes.
A table opens after an equals press on $λ$ : the equals key on the last box opens the table and stores the result in Ans. Tap the header's back arrow to return to the parameter boxes.

The fx-991EX screen in Poisson PD with x = 2, lambda = 3 and P = 0.2240418077.

Find a Poisson probability on your fx-991EX calculator

Enter x = 2 and a mean of 3 to find the chance of exactly two calls, then switch to Poisson CD for at most two.

Find the probability More guides

Find the probabilities of exactly two and at most two calls

Step 1

Step 2

Step 3

Step 4

Step 5

Read and check the result

If your result is different

Find a Poisson probability on your fx-991EX calculator