Expected Value Explained

What number should you expect if you roll a fair six-sided die a thousand times and average every roll? The answer keeps landing near 3.5, even though no single roll ever shows a 3.5. That steady average is expected value, the long-run average outcome of a random event. You find it by multiplying each possible result by how likely it is to happen, then adding those weighted results together.

What Is Expected Value?

Expected value is a way to summarize a random event with one number. It tells you the average result you would see if the event repeated many, many times. Statisticians call it EV for short.

Think of expected value as a weighted average. A plain average treats every outcome the same. Expected value instead gives more weight to outcomes that are more likely and less weight to outcomes that rarely happen.

You cannot find expected value from outcomes alone. You also need the probability attached to each one. That is the key input the formula in the next section relies on.

Expected value shows up in many everyday situations, including:

• Games of chance, like dice, cards, and lotteries
• Insurance, where companies estimate average payouts across many customers
• Business choices, like weighing possible profits against possible losses
• Simple bets, like a coin flip with different payouts

The Expected Value Formula

The formula for expected value is simple once you see it written out. Multiply each possible outcome by the probability of that outcome. Then add up all the results.

EV = (outcome 1 x probability 1) + (outcome 2 x probability 2) + (outcome 3 x probability 3) + …

Every probability in the formula must be a number between 0 and 1. All the probabilities for every possible outcome must also add up to exactly 1. That is what makes the weighting fair: no outcome gets more or less weight than its true chance of happening.

Worked Example: Rolling a Fair Six-Sided Die

A fair six-sided die is the classic expected value example, because every face is equally likely. Each of the six outcomes, 1 through 6, has a probability of 1/6.

You can also solve this the short way. Add the six faces: 1 + 2 + 3 + 4 + 5 + 6 = 21. Divide by 6 outcomes: 21 / 6 = 3.5. That matches the formula, since multiplying each face by 1/6 and adding the results gives the same 3.5.

Notice that 3.5 is not a number a die can ever show. It sits exactly between 3 and 4. That gap between the expected value and any real outcome is normal, and the next section explains why.

Worked Example: A Coin Flip With Different Payouts

Expected value gets more useful once outcomes are not equal. Picture a coin flip that pays $10 if it lands heads and $0 if it lands tails.

A fair coin gives each side a probability of 0.5. Multiply each payout by its probability: 0.5 x $10 = $5 for heads, and 0.5 x $0 = $0 for tails.

Add the two weighted results: $5.00 + $0.00 = $5.00. So the expected value of this coin flip is $5, even though any single flip pays either $10 or $0, never exactly $5.

Why Expected Value Is a Long-Run Average, Not a Guarantee

Expected value describes what happens on average across many repeats. It does not describe what happens on any one try. A single die roll or coin flip never shows the expected value itself.

The die roll example makes this clear. No face ever reads 3.5, yet 3.5 is still the correct expected value for one roll. Over many rolls, the average of all your results drifts closer to 3.5.

This idea has a name: the law of large numbers. It says the average result gets closer to the expected value as you repeat a random event more times. A few rolls can look uneven, but thousands of rolls smooth out.

Using Expected Value With Real Probabilities

Every expected value calculation depends on having the right probability for each outcome. A single six-sided die keeps this easy, since each face has the same 1/6 chance. Rolling two dice and looking at their sum is trickier, because a sum like 7 is more likely than a sum like 2.

A Dice Probability Calculator works out those probabilities for you, for one die or several dice at once. Once you have accurate probabilities, you can plug them straight into the expected value formula above.

This step matters because a wrong probability throws off the whole expected value. Getting the probability right first, then applying the formula, keeps the final number accurate.

How Expected Value Connects to Other Probability Ideas

Expected value builds on probability values, but it does not explain where those probabilities come from or how probability differs from odds. Our sibling guide, Probability vs Odds: What’s the Difference?, covers that distinction in full.

Expected value also does not involve counting arrangements or orderings, which is a separate skill covered in Combinations vs Permutations. Keep these ideas separate: expected value assumes you already know each outcome and its probability, not how to find or count them.

Can Expected Value Be Negative?

Yes. Expected value turns negative whenever the costs in a situation outweigh the average payout. This comes up constantly in games and raffles that charge an entry fee.

Picture a raffle ticket that costs $2. It pays $10 with a probability of 1/20, and $0 otherwise. The expected winnings are (1/20) x $10 + (19/20) x $0 = $0.50. Subtract the $2 cost, and the net expected value is $0.50 minus $2, which equals -$1.50.

A negative expected value does not mean you will lose every single time. It means that if you played the raffle many times, your average result would be a loss of $1.50 per ticket. That is exactly why lotteries and casino games are built with a negative expected value for the player, and a positive one for the organizer.

Common Mistakes When Calculating Expected Value

A few small errors trip people up when they calculate expected value by hand. Watch for these:

• Forgetting to convert a percent chance into a decimal before multiplying
• Leaving out a possible outcome, so the probabilities do not add up to 1
• Mixing up expected value with the single most likely outcome
• Rounding too early, which can throw off the final total

Checking that every probability adds up to 1 before you multiply catches most of these mistakes early.

Frequently Asked Questions About Expected Value

Sources

This article is for general education only. Statistical methods and examples use standard conventions and simplified data, so real-world analysis can be more complex. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 13, 2026.