Chips, fractions and percentages

Move between counts, fractions, decimals and percentages without changing the event.

8 minutes · beginner · Original Poker4Quant lesson

What you will learn

Convert a counted probability into equivalent forms and interpret percentage points.

One number, several representations

A probability of 1/4 is the decimal 0.25 and the percentage 25%. Divide favorable outcomes by total outcomes to get the decimal; multiply by 100 to write a percent. In a calculation, use 0.25 where the formula expects a probability, not 25.

A chip is an accounting unit here, not a cash prize. Keeping payoffs in chips avoids mixing currency conversion with the probability problem. If every chip amount doubles but the probabilities stay fixed, expected chip payoffs double; the probability itself does not.

Do not confuse units of change

Moving from 20% to 30% is an increase of 10 percentage points. Relative to the old probability, it is an increase of (30 − 20)/20 = 50%. These statements answer different questions.

Round the final displayed result, not each intermediate operation. For 1/3, keep the fraction or enough precision internally; 0.33 is an approximation, not exact equality. A precise decimal display also does not prove the input probabilities were accurately estimated.

Worked example

In a model with 40 equally likely outcomes, 10 win. A win nets 80 chips and a loss costs 20. Calculate the mean net payoff.

The win probability is 10/40 = 1/4 = 0.25.
The loss probability is 30/40 = 0.75.
Compute 0.25 × 80 − 0.75 × 20 = 20 − 15.

The expected net payoff is +5 chips. Using 25 instead of 0.25 would corrupt the units.

Common mistakes

Putting a percentage written as 25 directly into a formula that expects 0.25.
Reporting relative change when asked for percentage points.
Rounding each conditional probability before multiplying.

Keep these ideas

Fraction, decimal and percent can represent the same probability.
State chip and probability units explicitly.
Percentage-point changes differ from relative percent changes.