These questions ask you to count outcomes from two-stage experiments, predict how often an event should occur, or compare observed results with a fair model. Keep the model, the number of trials and the actual results as three separate pieces of information.
Build pairs, predict counts, then compare evidence
For two independent stages, pair every result from the first stage with every result from the second. If each of two fair spinners has three equal sections, there are 3 × 3 = 9 ordered pairs. A pair with first result A and second result B is different from first result B and second result A whenever the two stages are recorded separately. Once the sample space is complete, probability is favourable pairs divided by all pairs.
Expected frequency turns probability into an approximate count: multiply the event probability by the number of trials. For example, an event with probability 3/4 repeated 28 times has expected frequency 21. This is a long-run expectation, not a promise that every run gives exactly that count. Experimental probability instead uses what happened, such as 23 successes out of 30 trials. The theoretical probability stays tied to the fair device or stated model.
| Task | Method |
|---|---|
| Count a two-stage sample space | Multiply the number of possible results at the two stages and keep ordered pairs separate. |
| Find a probability from that space | Count favourable pairs out of all equally likely pairs. |
| Predict an event count | Multiply probability by number of trials and describe the result as about that many. |
| Judge fairness | Compare observed frequency with expected frequency, paying more attention to a large and persistent gap over many trials than to a small short-run difference. |
Where mistakes come from
| Tempting route | How to reject it |
|---|---|
| Add the numbers of results at two stages | Each first-stage result can pair with every second-stage result, so use multiplication to count pairs. |
| Merge a pair with its reversed pair | When stages are recorded separately, reversing the order produces a distinct outcome and must be counted again. |
| Use the number of faces or sections as the expected count | That count helps form the probability, but expected frequency also needs multiplication by the number of trials. |
| Replace the theoretical chance with the observed share | Observed results determine experimental probability. They do not rewrite the probability calculated from a fair model. |
| Declare a device fair merely because the target result exists | Fairness evidence comes from comparing observed and expected frequencies, not from checking whether a result is possible. |
| Trust a short run most, or think a missed result is now due | Larger experiments usually give steadier estimates. Independent trials do not remember earlier misses. |
Try it
Two fair coins are tossed together. What is the probability that both land on heads? Give the probability as a fraction out of the four equally likely outcomes.
- 1/4
- 2/4
- 1/3
- 3/4
A fair spinner has 5 equal sections, 2 of them red. The spinner is spun 100 times. About how many times would you expect it to land on red? Enter the number of red landings.
Two standard fair six-sided dice are rolled and the two scores are added. How many of the 36 equally likely outcomes give a total of 7, and what is the probability of that total as a fraction out of the 36 outcomes?
- 6 outcomes, a probability of 6/36
- 3 outcomes, a probability of 3/36, taking 1 with 6 and 6 with 1 as one case
- 7 outcomes, a probability of 7/36
- 1 outcome, a probability of 1/36, since 7 is a single total