7.3 Basic Probability
Key Takeaways
- OPM names basic probability as Occupational Math content, and it is the topic most often missing from unofficial USA Hire prep material.
- The probability of an event is the number of favourable outcomes divided by the total number of possible outcomes.
- Probability always lies between 0 and 1, so any result outside that range signals an error.
- The probability that an event does not occur is 1 minus the probability that it does.
- For independent events occurring together, multiply the probabilities; for mutually exclusive alternatives, add them.
7.3 Basic Probability
Quick Summary: OPM lists basic probability among the topics the Occupational Math Assessment covers. It is worth flagging that this topic is frequently absent from unofficial USA Hire prep material, which makes it a genuine gap for many candidates. The content required is elementary: the basic ratio definition, the complement rule, and combining events.
The basic definition
For equally likely outcomes:
Probability = favourable outcomes ÷ total possible outcomes
A tray holds 40 forms, of which 12 are marked for review. If one form is selected at random, what is the probability it is marked for review?
12 ÷ 40 = 0.3 = 3/10 = 30%.
Probability is always between 0 (impossible) and 1 (certain). Any value outside that range means something has gone wrong — usually an inverted division. Computing 40 ÷ 12 ≈ 3.33 is not a probability.
Three equivalent forms
| Form | Value |
|---|---|
| Fraction | 3/10 |
| Decimal | 0.3 |
| Percentage | 30% |
Match the form the options use. If they are percentages, convert (Section 5.3).
The complement rule
The probability that an event does not happen is 1 minus the probability that it does.
If the probability a form is marked for review is 0.3, the probability it is not marked is 1 − 0.3 = 0.7.
The complement is often the faster route. Questions phrased with at least one are the classic case.
Two forms are selected at random from a large batch in which 30% are marked. What is the probability at least one is marked?
Direct route: exactly one, plus exactly two — two calculations. Complement route: the opposite of "at least one" is "none". P(neither marked) = 0.7 × 0.7 = 0.49. So P(at least one) = 1 − 0.49 = 0.51.
One calculation instead of two, with less to go wrong. Whenever you see at least one, reach for the complement.
Combining events: and versus or
| Wording | Rule | When it applies |
|---|---|---|
| A and B both occur | Multiply | Events are independent |
| A or B occurs | Add | Events are mutually exclusive |
And — multiply
A batch is 30% marked. Two forms are drawn at random from a large batch. What is the probability both are marked?
0.30 × 0.30 = 0.09 = 9%.
Note that the combined probability is smaller than either individual probability. Requiring two things to happen together is harder than requiring one. If your "and" answer is larger than its components, you added when you should have multiplied.
Or — add
A tray holds 40 forms: 12 marked for review, 8 marked for filing, 20 unmarked. A form is drawn at random. What is the probability it is marked for review or for filing?
These categories cannot both apply, so add: 12/40 + 8/40 = 20/40 = 0.5 = 50%.
Adding only works when the categories are mutually exclusive. If a form could be marked both ways, simple addition would double-count the overlap.
Selection without replacement
When an item is removed and not replaced, the second probability changes because both the favourable count and the total shrink.
A tray holds 40 forms, 12 marked. Two are drawn without replacement. What is the probability both are marked?
First draw: 12/40 = 0.3. Second draw: now 11 marked remain out of 39 forms, so 11/39 ≈ 0.2821. Both: 0.3 × 0.2821 ≈ 0.0846, about 8.5%.
Compare with the with-replacement figure of 0.09. The difference is small here because the batch is reasonably large; with small batches it matters a great deal.
The signal words are without replacement, not returned, set aside, or simply a small stated population from which items are removed.
Counting outcomes
Some items need the total outcome count before a probability can be formed.
A code consists of one letter from A, B, C followed by one digit from 1 to 4. How many codes are possible, and what is the probability a randomly chosen code begins with B?
Total codes: 3 × 4 = 12. Codes beginning with B: 4 (B1, B2, B3, B4). Probability: 4 ÷ 12 = 1/3 ≈ 0.333.
When independent choices are made in sequence, multiply the number of options at each step.
A checklist
- Is the result between 0 and 1?
- Does the question say and (multiply) or or (add)?
- Is it with or without replacement?
- Would the complement be faster, particularly for at least one?
- Does my answer's form — fraction, decimal, or percentage — match the options?
A tray holds 40 forms, of which 12 are marked for review. Two forms are drawn without replacement. What is the probability that both are marked for review?
In a large batch, 30% of forms are marked. Two forms are drawn at random. What is the probability that at least one is marked?
A candidate computes a probability of 3.33 for drawing a marked form from a tray. What does this indicate?