Two-way tables
Use the total in the question: The question usually gives the total, e.g. "120 members".
Probability = how many you want ÷ the total. For example, 59 of the 120 members train in the morning, so P(morning) = 59/120.
| Morning | Evening | |
|---|---|---|
| Under 30 | 18 | 34 |
| 30 or over | 41 | 27 |
Read three kinds of probability
A gym asked 120 members when they train. The results are in the table above. A member is chosen at random.
(a) Find the probability that the member trains in the morning.
(b) Find the probability that the member is under 30 and trains in the evening.
(c) Find the probability that the member is not under 30.
Solution
- The total is in the question: 120 members. Check: the four cells add up to it.
- (a) Add the morning column.
- (b) One cell: where that row meets that column.
- (c) Add the other row.
Final answer
(a) 59/120 ≈ 0.492 (b) 34/120 ≈ 0.283 (c) 68/120 ≈ 0.567
Exam tip: Show the fraction, e.g. 59/120, before the decimal: it earns the method mark if a later number slips.
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Completing a two-way table
Fill the gap that is alone: Each row and each column adds to its total. Find a row or column with only ONE gap, fill it by subtracting, and repeat.
Fill the table from the words
A courier tracked 90 parcels. 30 of the parcels were large. 6 parcels were damaged, and 4 of the damaged parcels were large.
| Damaged | Not damaged | Total | |
|---|---|---|---|
| Large | |||
| Small | |||
| Total | 6 | 90 |
(a) Complete the table.
(b) A parcel is chosen at random. Find the probability that it is small and not damaged.
Solution
- Write in what the words give: large total 30, large and damaged 4.
- Large row: one gap. Not damaged = 30 − 4 = 26.
- Total column: small total = 90 − 30 = 60.
- Damaged column: small and damaged = 6 − 4 = 2.
- Small row: not damaged = 60 − 2 = 58. Check: 26 + 58 = 84 = 90 − 6.
- (b) One cell over the grand total.
Final answer
Large: 4, 26, 30. Small: 2, 58, 60. Totals: 6, 84, 90. (b) 58/90 ≈ 0.644.
| Damaged | Not damaged | Total | |
|---|---|---|---|
| Large | 4 | 26 | 30 |
| Small | 2 | 58 | 60 |
| Total | 6 | 84 | 90 |
Exam tip: Check the corner: the row totals and the column totals must both add to the grand total.
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Theoretical vs empirical probability
| Type | Definition |
|---|---|
| Theoretical | Based on logic (equally likely) |
| Empirical | Relative frequency from trials |
Interactive diagram
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Worked example
A seed company says that 5% of its seeds do not grow (theoretical). A gardener plants 400 seeds and 26 do not grow.
Find the experimental (relative frequency) probability that a seed does not grow.
Solution
- Relative frequency = 26/400 = 0.065
- More trials: the relative frequency settles near the true probability
Final answer
Experimental = 0.065 (claimed 0.05).
Expected number, and probability as a share of time
Expected number
3% of the bulbs a factory makes are faulty. Find the expected number of faulty bulbs in a batch of 2500.
Solution
- Multiply the number of bulbs by the probability for one bulb.
Final answer
75 faulty bulbs on average. An expected number is an average: it does not have to be a whole number.
Expected number: absent students
The probability that a student is absent on a given day is 0.1. Find the expected number of absent students in a class of 128.
Solution
- Multiply the number of students by the probability for one student.
Final answer
12.8 students. An expected number is an average over many days, so it does not have to be a whole number.
"At any instant" — use time: When something repeats in a cycle, the probability at a random instant is the share of one cycle it spends in that state: time in the state ÷ time for one cycle.
A share of the cycle
A lighthouse lamp turns at a constant rate, once every 5 seconds. Its beam shines on a boat for 1.2 seconds of every turn.
(a) At a random instant, find the probability that the beam is on the boat.
(b) The lamp now turns twice as fast. Find the probability now. Justify your answer.
Solution
- (a) Time in the beam ÷ time for one turn.
- (b) Twice as fast: 0.6 s in the beam out of a 2.5 s turn. Both times halve.
Final answer
(a) 0.24. (b) Still 0.24: the speed changes both times in the same ratio, so the share of each turn is unchanged.
Exam tip: A "Justify" needs the reason in words (the share of the cycle stays the same) AND the value. The value alone loses a mark.