Expected value versus realized results

A model can assign a decision an expected gain of 10 units while the next observation loses 50. Both numbers can be correct. Expected value averages the possible outcomes using their probabilities; a realized result records the outcome that occurred.

This lesson keeps the two records separate. Every fictional win gains 100 net units and every loss costs 50. You supply an assumed win probability and observed counts, then compare the model total with the recorded total. The exercise does not estimate a real hand’s winning chance or grade a decision.

Ace: “The expected result is 10, but neither outcome tray contains 10.”

Indira: “The trays contain the possible results, plus 100 and minus 50. Ten is their probability-weighted average under this model.”

Define the net outcomes first

The fixed outcomes are +100 for a win and -50 for a loss. They are net changes measured from the same decision point. The model has no ties, fees or later decisions, and those amounts stay the same on every trial.

Net accounting matters. Suppose calling costs 50 and the pot available before that call is 100. A win returns a final pot of 150, including the 50 you added. Your gain from the decision is 100. A loss costs the 50 call. Treating the full 150 as profit would mix gross receipts with a net loss.

Let p be the assumed win probability as a fraction. At 40%, p = 0.40 and the loss probability is 0.60. Multiply each net outcome by its probability: 0.40 x 100 + 0.60 x (-50) = 40 – 30 = 10 units.

For any supplied p, the same expression becomes 100p – 50(1 – p) = 150p – 50. The desk accepts a percentage, so entering 40 means 40%, not a probability of forty. It converts the percentage to a fraction for the calculation.

Keep probability and observed frequency separate

Use N for the number of observed trials and K for the number of wins. There are N – K losses. The realized total is 100K – 50(N – K), which simplifies to 150K – 50N.

The expected total is N x (150p – 50) when every trial has the supplied marginal win probability p and the same fixed outcomes. With p = 40% and N = 10, that total is 10 x 10 = 100 units.

Now enter four observed wins. The six losses leave a realized total of 4 x 100 – 6 x 50 = 100. Realized minus expected is 100 – 100 = 0. Equality in this example confirms the two totals match; it does not validate the probability assumption.

The observed win frequency is K/N. For four wins in ten trials, that is 40%. It happens to equal the model input here. If the observed count changes, the supplied probability stays at 40% until you deliberately edit it.

Estimating an unknown probability from observations is a separate statistical task. The sample may carry information, but its relevance depends on how it was collected and whether the conditions match. This desk performs no estimation and supplies no uncertainty interval.

Change the sample while holding the model fixed

Keep p = 40% and N = 10, then change K to six. Six wins contribute 600 and four losses cost 200. The realized total is 400, which is 300 above the expected total of 100.

Change K to two instead. Two wins contribute 200 and eight losses cost 400. The realized total is -200, which is 300 below expectation. The model still assigns +10 per trial because neither its probability nor its outcomes changed.

Each extra win replaces one -50 loss with one +100 gain. At a fixed trial count, the realized total therefore rises by 150 per additional win. Its difference from the unchanged expected total rises by the same amount.

Expected wins can also be fractional. At 40% over three trials, the expected count is 1.2 wins and the expected total is 30 units. An observed sample must have zero, one, two or three wins; its total is -150, 0, 150 or 300.

That three-trial example cannot realize exactly 30 under these fixed outcomes. An expectation need not be an attainable result of a particular sample. Rounding the expected win count to a whole number would change the calculation without adding any observation.

Ace: “I changed two observed wins to six. The result ledger moved, but the expected total stayed at 100.”

Indira: “Keep the 40% assumption and ten trials fixed. The observed tray changes the realized total, not the model probability.”

Review a winning sample under a losing model

Set the assumed win probability to 20% while keeping the same +100 and -50 outcomes. Expected value becomes 0.20 x 100 – 0.80 x 50 = -20 per trial. Ten trials have an expected total of -200.

If the supplied sample contains six wins, its realized total is still 400. It is now 600 above the model total. Editing the probability changed expectation; it did not change the six recorded wins or the net amounts assigned to them.

A decision review needs the information available before the outcome and the alternatives available then. A favorable river card does not prove an earlier call was best. A later loss also does not identify which alternative would have earned more.

Those alternative results are often unobserved. Folding, calling and raising can lead to different responses and future actions. Comparing them requires their own probabilities and payoffs; the result from the action actually taken does not fill in every missing branch.

Correct multiplication also leaves room for a poor model. An inaccurate opponent range, an omitted fee or a later bet can change expected value. Keep a check of the arithmetic separate from a check that the inputs describe the decision.

Read zero expectation and exact endpoints carefully

With these fixed outcomes, zero expected value occurs at p = 1/3: 150 x (1/3) – 50 = 0. This balances the probability-weighted gains and losses. Individual trials still gain 100 or lose 50.

The percentage input allows two decimal places, so it cannot express exactly one-third. Entering 33.33% gives EV = -0.005 per trial; 33.34% gives +0.01. A rounded 33.33% display should not be mistaken for the exact break-even fraction.

The endpoints have stronger implications. An exact p = 0% assigns zero probability to a win. If you enter any observed wins with that assumption, the observation and supplied model conflict. The desk should flag that conflict while retaining the observed count.

An exact p = 100% likewise conflicts with any observed loss. This does not mean a real recorded outcome must be denied. It means the exact probability assumption, the fixed-outcome description or the record needs review.

A probability close to an endpoint has a different meaning. At 0.01%, wins have positive marginal probability; at 99.99%, losses do. This desk does not calculate how unusual a sample is or determine which sequences a particular dependence model would permit.

Add expectations with the right assumptions

Expected totals add even when trials are dependent. If each trial has marginal win probability 40% and the same +100/-50 outcomes, each has expectation 10, so ten trials have expected total 100. Independence is not required for this addition.

Dependence matters for other questions, including which outcome patterns are possible and how earlier observations affect later probabilities. A shared average by itself does not supply a distribution of totals. The desk gives no standard deviation, confidence interval or forecast of a downswing.

If the probabilities differ, add the appropriate expectation for each trial. Five trials at 40% contribute 50 expected units. Five at 20% contribute -100. Together their expected total is -50. Treating all ten as 40% would incorrectly produce +100.

The same care applies to changing stakes or payoffs. Ten wins of different sizes cannot be represented by ten identical +100 entries. A real record with variable pots, ties, fees or later betting needs the actual outcomes and a model that includes them.

Ace: “I have the observed cards here and the probability assumption in a different file.”

Indira: “Keep both records. A mismatch tells us to inspect the model and the observations; it does not tell us which one to rewrite.”

Keep future expectations separate from the past result

Suppose the first ten trials at the 40% assumption recorded two wins and a total of -200. If the information now available still supports a 40% win probability for each of ten future trials, that next block has expected total +100.

The next block does not acquire an extra +300 expectation to repair the earlier gap. The recorded -200 is already known. Adding the future expectation gives a combined conditional expected total of -100 across the completed and future blocks.

This is different from the original, before-observation expectation of +200 for twenty trials under the unchanged marginal model. Once part of the outcome is known, it belongs in the calculation as the realized amount. The remaining uncertainty concerns the future part.

If past outcomes provide information that changes future probabilities, update those probabilities for a stated reason. The rule is to use the expectation supported by current information. A previous loss alone does not establish a schedule for recovery.

Use the expectation and result desk

Start with assumed win probability 40, trial count 10 and observed wins 4. Select Compare expectation and result. The model gives +10 per trial and +100 for ten trials; the observed total is +100 and realized minus expected is zero.

The probability input accepts 0 through 100 with at most two decimal places. Trials must be a whole number from 1 through 1,000,000. Observed wins must be a whole number from zero through the trial count; losses are the remaining trials.

Change only the observed wins to six, calculate, then try two. The expected total stays +100 while the realized totals become +400 and -200. Next keep the observations fixed and change the probability to 20% to see the model column change.

Editing a field clears stale output. Calculate again after each change. Reset example restores 40, 10 and 4. The desk has no live account connection, random simulation or decision-quality score; the worked tables remain readable without the controls.

A useful review note is specific: “With fixed net outcomes +100/-50 and assumed p = 40%, ten trials have expected total +100. Two observed wins produced -200, a difference of -300.” Keep the probability evidence beside that note. Related exercises are available in Info & Tools.

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