To calculate probability, first define the event and the possible outcomes. When individual outcomes are equally likely, divide the number that satisfy your event by the total number. For several events, the wording matters: “or” combines possibilities, “and” requires both, and “given” changes the group you are considering.
The examples below use an imaginary collection of colored tokens. The photographs illustrate the materials, not the exact counts in the calculations. You can follow everything with a pencil; the important work is deciding what belongs in the numerator and denominator.
Start with the sample space
A sample space is the complete set of possible outcomes. Suppose a bag holds 20 same-size tokens, 8 teal and 12 cream. Imagine each individual token has an equal chance of being selected. The sample space contains 20 individual tokens, even though there are only two color labels.
P(teal) = 8/20 = 2/5 = 0.4 = 40%. Writing P(teal) means “the probability of selecting teal.” The fraction, decimal, and percentage express the same chance. Our percentage calculation guide explains the conversion.
Do not divide by two just because two colors are available. The colors are not equally likely here. Likewise, counting objects is insufficient if their sizes, placement, or selection method give some a greater chance. OpenStax’s probability terminology makes the equal-likelihood condition explicit.
Use a two-way table to keep overlapping events straight
Now add a dot to some tokens without changing their selection chances. The following counts describe the entire imaginary collection. Every token belongs to one color row and one marking column.
| Color | Dotted | Plain | Total |
|---|---|---|---|
| Teal | 5 | 3 | 8 |
| Cream | 4 | 8 | 12 |
| Total | 9 | 11 | 20 |
The table answers several questions without a new formula. Dotted tokens account for 9/20. Teal and dotted tokens occupy the shared cell, giving 5/20, or 25%. Cream and plain tokens account for 8/20, or 40%. Check that row totals and column totals both reach 20 before using the table.
For “not,” subtract from one
The complement of an event includes every outcome outside it. Because teal and not teal exhaust this collection, P(not teal) = 1 − P(teal) = 1 − 2/5 = 3/5, or 60%. Here, “not teal” happens to mean cream. With more colors, it would include all the other colors together.
This approach is useful when the opposite event is easier to count. “At least one” is often easier to calculate by finding the chance of none, then subtracting from one. Be precise: the opposite of “both” is “not both,” which includes zero or one, rather than just zero.
For “or,” subtract any overlap
In probability, “A or B” normally includes outcomes satisfying either event or both. In our table, teal or dotted includes all 8 teal tokens and all 9 dotted tokens. But 5 tokens appear in both counts, so remove that duplicate count once.
P(teal or dotted) = 8/20 + 9/20 − 5/20 = 12/20 = 60%. You can check by counting the three included cells: 5 + 3 + 4 = 12.
The general rule is P(A or B) = P(A) + P(B) − P(A and B). If events cannot occur together, their overlap is zero. Teal and cream on one selection are mutually exclusive, whereas teal and dotted are not. OpenStax’s addition and multiplication rules explain both relationships.
For “given,” change the denominator
Suppose you learn that the selected token is dotted. The relevant group is now the 9 dotted tokens, not all 20. Five of those nine are teal, so P(teal given dotted) = 5/9, approximately 55.6%.
Reverse the question and the answer changes. Given that a token is teal, 5 of the 8 eligible tokens are dotted: P(dotted given teal) = 5/8, or 62.5%. Both questions use the same shared cell but different totals. Conditioning on an event with probability zero does not work with this division rule.
For “and,” check whether the first event changes the next
For two selections, multiply the first event’s probability by the second event’s probability given the first. With replacement, return the token and remix before selecting again. Under independent, equal-chance draws, the probability of two teal tokens is (8/20) × (8/20) = 4/25, or 16%.
Without replacement, a first teal token leaves 7 teal among 19 remaining tokens. The probability of two teal tokens becomes (8/20) × (7/19) = 14/95, approximately 14.74%. Keeping 8/20 for the second factor would ignore the changed collection.
For exactly one teal without replacement, there are two separate paths: teal then cream, or cream then teal. Add their probabilities: (8/20 × 12/19) + (12/20 × 8/19) = 48/95, approximately 50.53%. Neither path contains the other, so no overlap subtraction is needed.
Independence means learning one event occurred does not change the other’s probability. It does not mean events cannot occur together. See OpenStax’s explanation of independence and replacement for the distinction.
Separate calculated chance from observed frequency
Suppose you make 50 independent draws with replacement and record 23 teal results. The observed frequency is 23/50 = 46%. The model still assigns 40% to each draw. A short record need not match the theoretical chance exactly, and the next draw does not have to compensate for earlier results.
Before trusting an answer, check the eligible group, any overlap, and whether the collection changes between selections. Every probability must fall between zero and one. Keep fractions exact until the final decimal conversion.
To try the counting yourself, use matching objects you already own or browse plain wooden tokens on Amazon. A physical demonstration helps you see which outcomes belong together; it does not establish that your selection method is perfectly uniform.
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