Statistics & Probability · Topic 4 of 4
Probability & Risk
Theory
The Golden Rule: When a question involves two events (like rolling two dice or spinning two spinners), do not try to calculate the combinations in your head. You must draw a two-way grid/table to accurately count the total number of possible outcomes and the number of successful outcomes.
1. Simple Probability
Probability is a measure of how likely an event is to happen. It is calculated using the formula:
You can leave your answer as a fraction (simplified if possible), a decimal, or a percentage.
2. Dependent Events (Without Replacement)
If a question states that an object is drawn and "not replaced", the total number of items available for the next draw permanently decreases. You must also check if the specific "successful" items you are looking for were among those removed.
3. Expected Outcomes
You will frequently be asked to calculate how many times an event is expected to happen, and then compare it to the actual result.
- Step 1: Multiply the given probability by the total number of trials (e.g., if the probability is 0.05 and there are 400 trials, calculate ).
- Step 2: Compare your calculated "expected" value to the "actual" value given in the text, and clearly state whether the actual value is more or less than expected.
4. Comparing Probabilities
If asked to determine which of two different games offers a higher chance of winning, you must calculate the probability for both. Because fractions with different denominators (e.g., and ) are hard to compare directly, you should convert both fractions into decimals or percentages to make a valid, justified conclusion.
5. Combined Probability (in succession)
The probability of several independent events all happening is found by multiplying their probabilities. For example, if the probability of a train being on time is 0.7, the probability of three trains in a row all being on time is .
6. Bias & Reliability
A probability found from a survey or experiment is only trustworthy if it was gathered fairly. Two things can undermine it:
- Bias: the sample is not representative — surveying only one type of person skews the result (e.g. asking only cyclists whether a town needs a cycle lane).
- Small sample size: a small experiment may not reflect the true probability. The larger the sample, the closer the experimental probability gets to the theoretical one.
⚠️ Common Examiner Traps
- The "Lottery Bonus Ball" Trap: In a lottery draw where balls are not replaced, candidates frequently forget to subtract the drawn balls from the total. If 6 balls have been drawn from a machine of 49, the denominator for the 7th ball is 43, not 49. Furthermore, you must actively check if any of the balls already drawn fit the criteria for your next draw, as this reduces your numerator!
- The "Expected vs Actual" Mix-Up: When asked if an event happened "more or less than expected", candidates sometimes perform the calculation but fail to write the final conclusion sentence. If your expected value is 16.1, and the actual value was 15, you must write "Less than expected because 15 < 16.1" to secure the final mark.
- The "Invisible Table" Trap: For questions with two spinners, marks are awarded for identifying the correct total number of outcomes. If you do not draw a grid and accidentally miss a combination (thinking there are 15 outcomes instead of 16), you will lose the process marks.
- Adding instead of multiplying: for independent events happening in succession ("and then"), multiply the probabilities — do not add them.
Worked examples
Example 1
A lottery consists of a draw with 50 balls numbered from 1 to 50. In the draw, five numbered balls are drawn and not replaced. These five numbers were:
5, 12, 20, 31, 42
A further bonus ball is then drawn. Calculate the probability of the bonus ball being a multiple of 10. (2 marks)
Step 1: Identify the remaining total number of balls (the denominator). balls remaining.
Step 2: Identify how many "multiples of 10" existed originally. 10, 20, 30, 40, 50 (5 balls).
Step 3: Check the drawn list to see if any multiples of 10 have already been removed. The number 20 was drawn. Therefore, only 10, 30, 40, 50 remain (4 balls).
Final Answer: The probability is .
Example 2
Eddie runs a game stall at a school fayre. His game requires two spinners to be spun and allowed to come to rest.
- Spinner A: has numbers 2, 4, 6.
- Spinner B: has numbers 1, 3, 5, 7.
The numbers on which the spinners come to rest are multiplied together. To win a prize, the answer to this multiplication must be greater than 15. Calculate the probability of winning a prize. (3 marks)
Step 1: Draw a two-way grid to find all possible outcomes.
| 1 | 3 | 5 | 7 | |
|---|---|---|---|---|
| 2 | 2 | 6 | 10 | 14 |
| 4 | 4 | 12 | 20 | 28 |
| 6 | 6 | 18 | 30 | 42 |
Step 2: Count the total number of outcomes. There are 12 total outcomes.
Step 3: Count the successful outcomes (answers strictly greater than 15). The successful outcomes are 20, 28, 18, 30, and 42 (5 outcomes).
Final Answer: The probability of winning a prize is .
Example 3
A railway company runs a commuter service. The probability of a train arriving late is 0.045.
In one month, the company ran 600 trains, of which 24 arrived late.
Determine if the number of late trains is more or less than expected. (2 marks)
Step 1: Calculate the expected number of late trains by multiplying the probability by the total number of trains.
Step 2: Compare the expected value (27) to the actual value (24) and state a conclusion.
Final Answer: Less than expected, because 24 is less than 27.
Example 4
Combined Probability (in succession)
At a station, the probability that any train arrives on time is 0.8. Assuming the trains are independent, calculate the probability that the next three trains all arrive on time. Round your answer to 2 decimal places.
Step 1: The three arrivals are independent events happening in succession, so multiply the probabilities:
Step 2: Evaluate:
Final Answer: The probability is (to 2 d.p.).
Example 5
🎯 Exam-style (bias & reliability)
A town council wants to find out whether residents support building a new cycle lane. A researcher surveys 40 people as they leave a bicycle shop, and finds that 90% are in favour.
(a) Explain why this result is likely to be biased. (b) Suggest one way to make the survey more reliable.
(a) People leaving a bicycle shop are far more likely to be cyclists, who would tend to support a cycle lane. The sample is not representative of all residents, so it is biased towards being in favour and over-states the true level of support.
(b) Survey a larger, random cross-section of residents from different places around the town (not just cyclists) — a bigger, representative sample gives a more reliable result.