Statistics expected value and probability distributions worksheet

Multiply each outcome by its probability and add. The result is the long-run average per trial, not a value that has to occur. For a game, subtract the cost of playing to get the expected profit; a negative answer means the house wins over time.

Select your difficulty

Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Expected Value and Probability Distributions Date Period Answer each question. $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 18 & 1 & 9 & 12 \\ P (x) & \frac{1}{12} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \end{array} \end{aligned}$ A game costs $$ 4.00$ to play and pays $$ 17.52$ with probability $\frac{1}{2}$ . Find the expected profit $\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{3}{8} & ? & \frac{1}{8} & \frac{3}{8} \end{array} \end{aligned}$ $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & - 1 & - 4 & 5 & 9 \\ P (x) & \frac{1}{4} & \frac{1}{2} & \frac{1}{8} & \frac{1}{8} \end{array} \end{aligned}$ A game costs $$ 5.00$ to play and pays $$ 36.20$ with probability $\frac{3}{20}$ . Find the expected profit $\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{7}{12} & ? & \frac{1}{4} & \frac{1}{12} \end{array} \end{aligned}$ $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & - 2 & 2 & 4 & 0 \\ P (x) & \frac{13}{16} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} \end{array} \end{aligned}$ A game costs $$ 4.00$ to play and pays $$ 14.22$ with probability $\frac{1}{2}$ . Find the expected profit $\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & ? & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} \end{array} \end{aligned}$ $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 11 & 17 & 15 & - 3 \\ P (x) & \frac{5}{8} & \frac{1}{8} & \frac{1}{16} & \frac{3}{16} \end{array} \end{aligned}$ A game costs $$ 5.00$ to play and pays $$ 11.70$ with probability $\frac{1}{2}$ . Find the expected profit $\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{3}{4} & \frac{1}{12} & ? & \frac{1}{12} \end{array} \end{aligned}$ $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 4 & - 3 & 2 & 10 \\ P (x) & \frac{1}{8} & \frac{3}{4} & \frac{1}{16} & \frac{1}{16} \end{array} \end{aligned}$ A game costs $$ 5.00$ to play and pays $$ 16.20$ with probability $\frac{3}{10}$ . Find the expected profit $\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{1}{4} & \frac{1}{8} & \frac{3}{16} & ? \end{array} \end{aligned}$ $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 2 & - 3 & - 2 & - 4 \\ P (x) & \frac{1}{8} & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} \end{array} \end{aligned}$ A game costs $$ 4.00$ to play and pays $$ 19.84$ with probability $\frac{3}{8}$ . Find the expected profit $\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{5}{12} & ? & \frac{1}{3} & \frac{1}{12} \end{array} \end{aligned}$ $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & - 1 & - 5 & 18 & 2 \\ P (x) & \frac{13}{16} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} \end{array} \end{aligned}$ A game costs $$ 4.00$ to play and pays $$ 16.50$ with probability $\frac{1}{2}$ . Find the expected profit Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Expected Value and Probability Distributions

Date Period

Answer each question.

$\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 18 & 1 & 9 & 12 \\ P (x) & \frac{1}{12} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \end{array} \end{aligned}$
A game costs $$ 4.00$ to play and pays $$ 17.52$ with probability $\frac{1}{2}$ . Find the expected profit
$\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{3}{8} & ? & \frac{1}{8} & \frac{3}{8} \end{array} \end{aligned}$
$\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & - 1 & - 4 & 5 & 9 \\ P (x) & \frac{1}{4} & \frac{1}{2} & \frac{1}{8} & \frac{1}{8} \end{array} \end{aligned}$
A game costs $$ 5.00$ to play and pays $$ 36.20$ with probability $\frac{3}{20}$ . Find the expected profit
$\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{7}{12} & ? & \frac{1}{4} & \frac{1}{12} \end{array} \end{aligned}$
$\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & - 2 & 2 & 4 & 0 \\ P (x) & \frac{13}{16} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} \end{array} \end{aligned}$
A game costs $$ 4.00$ to play and pays $$ 14.22$ with probability $\frac{1}{2}$ . Find the expected profit
$\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & ? & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} \end{array} \end{aligned}$
$\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 11 & 17 & 15 & - 3 \\ P (x) & \frac{5}{8} & \frac{1}{8} & \frac{1}{16} & \frac{3}{16} \end{array} \end{aligned}$
A game costs $$ 5.00$ to play and pays $$ 11.70$ with probability $\frac{1}{2}$ . Find the expected profit
$\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{3}{4} & \frac{1}{12} & ? & \frac{1}{12} \end{array} \end{aligned}$
$\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 4 & - 3 & 2 & 10 \\ P (x) & \frac{1}{8} & \frac{3}{4} & \frac{1}{16} & \frac{1}{16} \end{array} \end{aligned}$
A game costs $$ 5.00$ to play and pays $$ 16.20$ with probability $\frac{3}{10}$ . Find the expected profit
$\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{1}{4} & \frac{1}{8} & \frac{3}{16} & ? \end{array} \end{aligned}$
$\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 2 & - 3 & - 2 & - 4 \\ P (x) & \frac{1}{8} & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} \end{array} \end{aligned}$
A game costs $$ 4.00$ to play and pays $$ 19.84$ with probability $\frac{3}{8}$ . Find the expected profit
$\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{5}{12} & ? & \frac{1}{3} & \frac{1}{12} \end{array} \end{aligned}$
$\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & - 1 & - 5 & 18 & 2 \\ P (x) & \frac{13}{16} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} \end{array} \end{aligned}$
A game costs $$ 4.00$ to play and pays $$ 16.50$ with probability $\frac{1}{2}$ . Find the expected profit



Sigma Prep · sigmaprep.io/worksheets Name Expected Value and Probability DistributionsAnswer keyVersion 1 Date Period Answer each question. $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 18 & 1 & 9 & 12 \\ P (x) & \frac{1}{12} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \end{array} \end{aligned}$ $\frac{20}{3}$ A game costs $$ 4.00$ to play and pays $$ 17.52$ with probability $\frac{1}{2}$ . Find the expected profit $$ 4.76$ $\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{3}{8} & ? & \frac{1}{8} & \frac{3}{8} \end{array} \end{aligned}$ $\frac{1}{8}$ $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & - 1 & - 4 & 5 & 9 \\ P (x) & \frac{1}{4} & \frac{1}{2} & \frac{1}{8} & \frac{1}{8} \end{array} \end{aligned}$ $- \frac{1}{2}$ A game costs $$ 5.00$ to play and pays $$ 36.20$ with probability $\frac{3}{20}$ . Find the expected profit $$ 0.43$ $\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{7}{12} & ? & \frac{1}{4} & \frac{1}{12} \end{array} \end{aligned}$ $\frac{1}{12}$ $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & - 2 & 2 & 4 & 0 \\ P (x) & \frac{13}{16} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} \end{array} \end{aligned}$ $- \frac{5}{4}$ A game costs $$ 4.00$ to play and pays $$ 14.22$ with probability $\frac{1}{2}$ . Find the expected profit $$ 3.11$ $\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & ? & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} \end{array} \end{aligned}$ $\frac{1}{8}$ $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 11 & 17 & 15 & - 3 \\ P (x) & \frac{5}{8} & \frac{1}{8} & \frac{1}{16} & \frac{3}{16} \end{array} \end{aligned}$ $\frac{75}{8}$ A game costs $$ 5.00$ to play and pays $$ 11.70$ with probability $\frac{1}{2}$ . Find the expected profit $$ 0.85$ $\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{3}{4} & \frac{1}{12} & ? & \frac{1}{12} \end{array} \end{aligned}$ $\frac{1}{12}$ $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 4 & - 3 & 2 & 10 \\ P (x) & \frac{1}{8} & \frac{3}{4} & \frac{1}{16} & \frac{1}{16} \end{array} \end{aligned}$ $- 1$ A game costs $$ 5.00$ to play and pays $$ 16.20$ with probability $\frac{3}{10}$ . Find the expected profit $- $ 0.14$ $\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{1}{4} & \frac{1}{8} & \frac{3}{16} & ? \end{array} \end{aligned}$ $\frac{7}{16}$ $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 2 & - 3 & - 2 & - 4 \\ P (x) & \frac{1}{8} & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} \end{array} \end{aligned}$ $- \frac{9}{4}$ A game costs $$ 4.00$ to play and pays $$ 19.84$ with probability $\frac{3}{8}$ . Find the expected profit $$ 3.44$ $\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{5}{12} & ? & \frac{1}{3} & \frac{1}{12} \end{array} \end{aligned}$ $\frac{1}{6}$ $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & - 1 & - 5 & 18 & 2 \\ P (x) & \frac{13}{16} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} \end{array} \end{aligned}$ $\frac{1}{8}$ A game costs $$ 4.00$ to play and pays $$ 16.50$ with probability $\frac{1}{2}$ . Find the expected profit $$ 4.25$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Expected Value and Probability DistributionsAnswer keyVersion 1

Date Period

Answer each question.

$\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 18 & 1 & 9 & 12 \\ P (x) & \frac{1}{12} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \end{array} \end{aligned}$ $\frac{20}{3}$
A game costs $$ 4.00$ to play and pays $$ 17.52$ with probability $\frac{1}{2}$ . Find the expected profit $$ 4.76$
$\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{3}{8} & ? & \frac{1}{8} & \frac{3}{8} \end{array} \end{aligned}$ $\frac{1}{8}$
$\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & - 1 & - 4 & 5 & 9 \\ P (x) & \frac{1}{4} & \frac{1}{2} & \frac{1}{8} & \frac{1}{8} \end{array} \end{aligned}$ $- \frac{1}{2}$
A game costs $$ 5.00$ to play and pays $$ 36.20$ with probability $\frac{3}{20}$ . Find the expected profit $$ 0.43$
$\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{7}{12} & ? & \frac{1}{4} & \frac{1}{12} \end{array} \end{aligned}$ $\frac{1}{12}$
$\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & - 2 & 2 & 4 & 0 \\ P (x) & \frac{13}{16} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} \end{array} \end{aligned}$ $- \frac{5}{4}$
A game costs $$ 4.00$ to play and pays $$ 14.22$ with probability $\frac{1}{2}$ . Find the expected profit $$ 3.11$
$\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & ? & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} \end{array} \end{aligned}$ $\frac{1}{8}$
$\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 11 & 17 & 15 & - 3 \\ P (x) & \frac{5}{8} & \frac{1}{8} & \frac{1}{16} & \frac{3}{16} \end{array} \end{aligned}$ $\frac{75}{8}$
A game costs $$ 5.00$ to play and pays $$ 11.70$ with probability $\frac{1}{2}$ . Find the expected profit $$ 0.85$
$\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{3}{4} & \frac{1}{12} & ? & \frac{1}{12} \end{array} \end{aligned}$ $\frac{1}{12}$
$\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 4 & - 3 & 2 & 10 \\ P (x) & \frac{1}{8} & \frac{3}{4} & \frac{1}{16} & \frac{1}{16} \end{array} \end{aligned}$ $- 1$
A game costs $$ 5.00$ to play and pays $$ 16.20$ with probability $\frac{3}{10}$ . Find the expected profit $- $ 0.14$
$\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{1}{4} & \frac{1}{8} & \frac{3}{16} & ? \end{array} \end{aligned}$ $\frac{7}{16}$
$\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & 2 & - 3 & - 2 & - 4 \\ P (x) & \frac{1}{8} & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} \end{array} \end{aligned}$ $- \frac{9}{4}$
A game costs $$ 4.00$ to play and pays $$ 19.84$ with probability $\frac{3}{8}$ . Find the expected profit $$ 3.44$
$\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{ccccc} outcome & A & B & C & D \\ P & \frac{5}{12} & ? & \frac{1}{3} & \frac{1}{12} \end{array} \end{aligned}$ $\frac{1}{6}$
$\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{ccccc} x & - 1 & - 5 & 18 & 2 \\ P (x) & \frac{13}{16} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} \end{array} \end{aligned}$ $\frac{1}{8}$
A game costs $$ 4.00$ to play and pays $$ 16.50$ with probability $\frac{1}{2}$ . Find the expected profit $$ 4.25$

How to do these

A game costs $$ 2.00$ to play and pays $$ 10.00$ with probability $\frac{1}{8}$ . Find the expected profit

Expected winnings: 10 times one eighth, which is 1.25.
Take away the 2.00 to play.

The answer is $- $ 0.75$ .

Where students go wrong

Forgetting the cost, or forgetting that the probabilities have to add to 1 before a missing one can be found.

Also called probability distributions, expected payoff, fair games or discrete random variables.

What you can put on this worksheet

Expected value from a table: $\begin{aligned} e x t F i n d t h e e x p e c t e d v a l u e \\ \begin{array}{cccc} x & 1 & 3 & 4 \\ P (x) & \frac{2}{5} & \frac{2}{5} & \frac{1}{5} \end{array} \end{aligned}$ → $\frac{12}{5}$
Expected profit of a game: A game costs $$ 1.00$ to play and pays $$ 6.05$ with probability $\frac{2}{5}$ . Find the expected profit → $$ 1.42$
The missing probability: $\begin{aligned} e x t F i n d t h e m i s s i n g p r o b a b i l i t y \\ \begin{array}{cccc} outcome & A & B & C \\ P & ? & \frac{2}{5} & \frac{1}{5} \end{array} \end{aligned}$ → $\frac{2}{5}$

Questions about these worksheets

Yes. Every expected value and probability distributions sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Statistics is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the expected value and probability distributions questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: expected value from a table, expected profit of a game and the missing probability. Tick as many as you want and set how many of each, or let it spread them evenly.

Forgetting the cost, or forgetting that the probabilities have to add to 1 before a missing one can be found.

Statistics expected value and probability distributions worksheet

Select your difficulty

Expected Value and Probability Distributions

Expected Value and Probability DistributionsAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

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