Case Opening Odds and Expected Value: A Worked Example

Collect the full probability schedule first

A usable calculation needs the full opening cost, every possible outcome, its probability and a value measured on a consistent basis. NIST's description of a discrete distribution requires nonnegative probabilities that sum to one. For a percentage schedule, that means 100% across all mutually exclusive outcomes. Equal space in a prize gallery does not supply those probabilities.

Keep the case version and observation time alongside the schedule. If probabilities cover broad groups, you also need the value distribution within each group; assigning every item in a group the same value is an assumption. Do not silently fill missing probabilities with equal shares or substitute another case's odds. If displayed rounding prevents an exact total, identify that precision limit instead of presenting a falsely exact result.

[1] NIST/SEMATECH e-Handbook: What is a Probability Distribution?

A fictional USD 10 case returns USD 7.50 on average

Assume an entirely fictional opening costs USD 10 and returns exactly one of four outcomes. Their assigned values are USD 1 with 80% probability, USD 5 with 10%, USD 30 with 9%, and USD 350 with 1%. These probabilities total 100%. They are invented teaching inputs, not Valve drop rates, operator odds or current skin prices.

Multiply each value by its probability expressed as a decimal. The contributions are USD 1 multiplied by 0.80 = USD 0.80; USD 5 multiplied by 0.10 = USD 0.50; USD 30 multiplied by 0.09 = USD 2.70; and USD 350 multiplied by 0.01 = USD 3.50. Add them: the expected gross assigned value is USD 7.50.

Subtract the USD 10 cost once to get an expected net result of minus USD 2.50 per opening. Dividing USD 7.50 by USD 10 gives a theoretical return of 75%; the corresponding shortfall is 25% of cost. This follows the probability-weighted payout method illustrated in the Gambling Commission's machine example, but the figures here are original and unrelated to an actual product.

An unweighted average would be USD 96.50: the sum of USD 1, USD 5, USD 30 and USD 350 divided by four. That answer is wrong for this schedule because it treats the rarest outcome as equally likely as the commonest. The weights carry the information that the attractive prize images leave out.

[1] NIST/SEMATECH e-Handbook: What is a Probability Distribution?[2] Gambling Commission: How can gaming machines meet their %RTP if they are random?

A 75% return is not a 75% chance of profit

In the fictional schedule, both the USD 1 and USD 5 outcomes return less than the USD 10 cost. Together they account for 90% of openings. The USD 30 and USD 350 outcomes exceed cost and together account for 10%. There is no break-even outcome. Receiving an item every time therefore coexists with a 90% probability of a net loss on a single opening.

The most likely outcome is USD 1, with an 80% probability; USD 7.50 is not an available prize at all. The Gambling Commission explains that RTP describes an average across extensive play, not what an individual session must return. Its guidance concerns gaming machines in Great Britain. We use the distinction to interpret the fictional model, not to claim that a named case site meets British licensing or testing requirements.

[1] NIST/SEMATECH e-Handbook: What is a Probability Distribution?[3] Gambling Commission: Return to player: how much gaming machines payout

Why 1% does not mean one prize every 100 attempts

For this separate calculation, assume successive fictional openings are independent and the USD 350 outcome keeps its 1% probability. The chance of missing it once is 99%. The chance of missing it in all 100 openings is 0.99 raised to the power of 100, approximately 36.60%. Consequently, the chance of at least one is approximately 63.40%, not 100%.

NIST's binomial model gives an expected count of 100 multiplied by 0.01, or one, but an expected count is not an appointment. With independence, earlier misses do not increase the next opening's 1% chance. These calculations do not establish independence for any operator and are not a suggested number of attempts. Repetition adds cost; it does not make the negative expected result disappear.

[4] NIST/SEMATECH e-Handbook: Binomial Distribution[5] NIST Computer Security Resource Center: Binomial Distribution

Recalculate when assigned value differs from realizable cash

Label the valuation basis before calling the result cash return. Steam's Community Market FAQ says wallet sale proceeds cannot be withdrawn to a bank. A Steam reference price therefore cannot establish what an external buyer will actually pay. Until resale amounts are supported, the USD 7.50 figure describes only the fictional assigned values.

As another explicit assumption, reduce every fictional prize's value by 10% to represent a uniform total resale deduction, with no other charges. Expected net proceeds become USD 7.50 multiplied by 0.90 = USD 6.75. After the unchanged USD 10 opening cost, expected loss becomes USD 3.25. Real item-specific costs would need to be applied separately to each outcome. This sensitivity calculation illustrates a missing input; it does not estimate an actual cashout rate.

[6] Steam Support: Community Market FAQ[2] Gambling Commission: How can gaming machines meet their %RTP if they are random?

Audit the claim without paying for a sample

Before accepting an expected-value claim, check whether it includes all outcomes, uses the relevant cost, dates its valuations and distinguishes theoretical calculations from recorded results. The Gambling Commission's monitoring guidance separates designed RTP from actual RTP and explains why volatility affects interpretation. A handful of selected openings cannot establish the accuracy of an entire probability schedule.

Write unknown beside unavailable inputs. A reproducible calculation using unsupported prices is still conditional, and a favorable screenshot does not prove the next result. If the question is what the model implies, the arithmetic can be checked without opening a case. This example identifies expected loss and uncertainty; it supplies no profitable strategy or reason to continue spending.

[7] Gambling Commission: Live return to player performance monitoring of games of chance