These questions ask for the chance of one result, a group of results, or everything outside a stated event. Solve them by counting outcomes, then decide whether direct counting or subtraction from 1 is quicker.
Count the whole and the favourable part
First confirm that the outcomes are equally likely. Count everything that could happen for the denominator, and count every outcome meeting the condition for the numerator. If a jar holds 4 orange, 2 purple and 3 white beads, the chance of orange or purple is 6/9: those two colours contribute six favourable beads, while all nine beads remain in the denominator. The names or numbers printed on outcomes do not themselves tell you how many outcomes qualify.
For a complement, imagine splitting the whole sample space into A and not A. If P(A) = 3/8, then P(not A) = 1 - 3/8 = 5/8. You can also count the five outcomes outside A directly. With decimals, subtract from 1 in the same units: if an event has probability 0.42, its complement has probability 0.58.
| Question wording | Method |
|---|---|
| One stated result or several stated results | Count every favourable outcome and divide by the total number of equally likely outcomes. |
| Not, does not, or avoids | Count everything outside the event, or subtract the event probability from 1. |
| Two disjoint categories joined by or | Combine their favourable counts while keeping the same total sample space. |
Where mistakes come from
| Tempting route | How to reject it |
|---|---|
| Use a printed number as the numerator | A label identifies an outcome; it does not count favourable outcomes. Count how many outcomes meet the condition. |
| Use the size of another group as the denominator | The denominator is the whole set of possible outcomes, including both favourable and unfavourable ones. |
| Treat a property as one outcome | A word such as even can describe several separate results. Check every possible outcome against the condition. |
| Repeat the event probability when asked for not the event | The wording has switched to the complementary part. The two probabilities must add to 1. |
| Change the denominator instead of subtracting a fraction from a whole | Keep the same equal parts. Subtract the numerator from the whole numerator while preserving the denominator. |
| Add denominators when combining categories | The sample space has not doubled. Add favourable counts over the original total. |
Try it
A fair coin is tossed once. What is the probability that it lands on tails? Give the probability as a fraction out of the two equally likely outcomes.
- 1/1
- 2/1
- 1/4
- 1/2
A fair spinner has 10 equal sections: 4 red, 3 blue and 3 green. How many of the sections are not blue? Enter the number of sections.
A weather forecast for one town states that the probability of rain tomorrow is 0.3. Taking that figure as given, what is the probability that it does not rain tomorrow there? Enter the probability as a decimal.