Cambridge Lower Secondary CheckpointStage 9

Statistics and Probability (9Ss)

Mathematics Stage 9 Chapter Notes

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Statistics and Probability - Statistics

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Statistics and Probability (9Ss) notes

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In this chapter

1Mutually Exclusive Events
2Experimental Probability (Relative Frequency)
3Calculating Expected Frequency
4Two-Way Tables and Probability
5Probability of Independent Events
6Tree Diagrams for Combined Events

1. Mutually Exclusive Events

Mutually exclusive events are events that cannot happen at the same time. For example, when you roll a standard six-sided dice, you can roll a 3 or a 4, but you cannot roll a 3 and a 4 in a single throw. The outcomes are mutually exclusive. A crucial rule for this type of event is that the sum of the probabilities of all possible mutually exclusive outcomes is always equal to 1. This is because one of the outcomes must occur.

For mutually exclusive events A and B: P(A or B) = P(A) + P(B)

Sum of P(all mutually exclusive outcomes) = 1

Key term

Mutually Exclusive: Events that cannot occur at the same time.

Examiner insight

Examiners often use algebra in these questions. Be prepared to form and solve a simple linear equation where the sum of expressions for probabilities equals 1.

Common pitfall

Forgetting that the probabilities of ALL possible outcomes must be included to sum to 1. If an outcome is not mentioned, you may need to calculate its probability first.

Worked example 12 marks

A bag contains only red, blue, and green counters. A counter is picked at random. The probability of picking a red counter is 0.35 and the probability of picking a blue counter is 0.2. What is the probability of picking a green counter?

1
The events 'picking red', 'picking blue', and 'picking green' are mutually exclusive. Their probabilities must sum to 1.
2
Let P(R) = 0.35, P(B) = 0.2, and P(G) be the probability of picking a green counter.
3
P(R) + P(B) + P(G) = 1
4
0.35 + 0.2 + P(G) = 1
5
0.55 + P(G) = 1
6
P(G) = 1 - 0.55
7
P(G) = 0.45

Worked example 23 marks

A biased spinner can land on A, B, C, or D. The probabilities of landing on A, B, and C are shown in the table. The probability of landing on D is twice the probability of landing on A. Find the probability of landing on B.

1
Let the probability of landing on B be x. So, P(B) = x.
2
The probability of landing on D is twice that of A: P(D) = 2 × P(A) = 2 × 0.15 = 0.30.
3
The sum of all probabilities must be 1: P(A) + P(B) + P(C) + P(D) = 1.
4
Substitute the known values: 0.15 + x + 0.4 + 0.30 = 1.
5
Combine the known probabilities: 0.85 + x = 1.
6
Solve for x: x = 1 - 0.85 = 0.15.
7
So, the probability of landing on B is 0.15.

Recap

Mutually exclusive events cannot happen together.
The probabilities of all possible mutually exclusive outcomes add up to 1.
To find a missing probability, subtract the sum of the known probabilities from 1.
The 'OR' rule for mutually exclusive events means you add their probabilities: P(A or B) = P(A) + P(B).

Quick check

The probability of a train being early is 0.1 and late is 0.4. If it can only be early, late or on time, what is the probability it is on time?1 mark

2. Experimental Probability (Relative Frequency)

While theoretical probability tells you what should happen in theory (e.g., P(Heads) = 0.5), experimental probability is what actually happens when you conduct an experiment. It's calculated as the relative frequency of an outcome. The more trials you conduct in an experiment, the closer your experimental probability is likely to get to the theoretical probability. For example, if you flip a coin 10 times, you might get 7 heads (relative frequency = 0.7), but if you flip it 1000 times, you're likely to get much closer to 500 heads (relative frequency ≈ 0.5).

Relative Frequency = (Number of times an outcome occurs) / (Total number of trials)

Key term

Relative Frequency: An estimate of probability based on the results of an experiment or observation.

Examiner insight

Examiners look for your ability to use experimental data to make judgements. A common question asks you to compare a relative frequency to a theoretical probability and comment on whether an object (like a coin or dice) is fair.

Common pitfall

Confusing frequency with relative frequency. Frequency is a count, while relative frequency is a proportion (fraction, decimal, or percentage).

Fun fact

Insurance companies rely heavily on experimental probability (using historical data) to set your premiums. The number of past claims in your area for your age group helps them estimate the probability you'll make a claim.

Worked example 13 marks

A biased dice is rolled 200 times. The results are recorded in the table.

Score	1	2	3	4	5	6
Frequency	34	28	50	31	27	30

a) What is the relative frequency of rolling a 3?b) What is the relative frequency of rolling an even number?

1
a) The number of times a 3 was rolled is 50. The total number of trials is 200.
2
Relative Frequency of 3 = 50 / 200
3
Simplify the fraction: 50/200 = 1/4 or 0.25.
4
b) First, find the total frequency of even numbers (2, 4, 6).
5
Frequency of evens = F(2) + F(4) + F(6) = 28 + 31 + 30 = 89.
6
The total number of trials is 200.
7
Relative Frequency of even = 89 / 200.

Recap

Experimental probability is found by doing an experiment.
Relative frequency is the fraction or decimal representing the experimental probability.
The formula is: (Number of successes) / (Total trials).
More trials generally lead to a more reliable estimate of the true probability.
An experimental probability that is very different from the theoretical probability may suggest bias.

Quick check

A spinner is spun 50 times and lands on blue 15 times. What is the relative frequency of landing on blue?1 mark

3. Calculating Expected Frequency

Expected frequency is a prediction. It's the number of times you would expect an outcome to occur over a certain number of trials. It bridges the gap between theoretical probability and real-world experiments. If you know the probability of an event, you can calculate how many times you'd expect it to happen. For example, if the probability of winning a game is 1/10, and you play 50 times, you would expect to win about 5 times.

Expected Frequency = P(event) × Number of trials

Key term

Expected Frequency: The number of times an outcome is predicted to occur in a specific number of trials.

Examiner insight

Ensure your answer makes sense in the context. If you are calculating the expected number of people, your answer should be a whole number if the context implies it, though often a decimal answer is acceptable as it is an average expectation.

Common pitfall

Using an observed frequency from an experiment instead of the theoretical probability in the formula. Expected frequency is based on what *should* happen, not what *did* happen in one specific experiment.

Worked example 12 marks

A fair six-sided dice is rolled 300 times. How many times would you expect to roll a number greater than 4?

1
First, identify the event and its probability. The numbers greater than 4 are 5 and 6.
2
There are 2 successful outcomes out of 6 possible outcomes.
3
P(greater than 4) = 2 / 6 = 1 / 3.
4
Now, use the expected frequency formula. The number of trials is 300.
5
Expected Frequency = P(event) × Number of trials
6
Expected Frequency = (1/3) × 300
7
Expected Frequency = 100.
8
You would expect to roll a number greater than 4 about 100 times.

Worked example 22 marks

The probability that a factory component is faulty is 0.02. In a production run of 5000 components, how many would you expect to be faulty?

1
Identify the probability and the number of trials.
2
P(faulty) = 0.02
3
Number of trials = 5000
4
Use the formula: Expected Frequency = P(event) × Number of trials
5
Expected Frequency = 0.02 × 5000
6
Expected Frequency = 100.
7
You would expect 100 components to be faulty.

Recap

Expected frequency is a prediction of how often an outcome will occur.
The formula is: Probability of the event multiplied by the number of trials.
You must know the theoretical probability of an event to calculate its expected frequency.
The answer is an expectation, not a guarantee; the actual result may be different.

Quick check

The probability of rain on any given day in April is 0.4. How many rainy days would you expect in the 30 days of April?1 mark

4. Two-Way Tables and Probability

Two-way tables are a powerful tool for organising and displaying data from two different categories. For example, you could survey students on their gender and their preferred mode of transport. The table allows you to see the frequencies for each combination (e.g., number of boys who walk to school). Once the table is complete, you can easily calculate probabilities by using the totals. The number in the bottom-right corner is the grand total, which is usually the denominator for your probability calculations.

P(event) = (Number of items in event category) / (Grand Total)

Key term

Two-Way Table: A table used to display frequencies for two different categories of data.

Examiner insight

Marks are often awarded for correctly completing the table before any probability is calculated. Show your completed table clearly in your working.

Common pitfall

Using the wrong total for the denominator. Unless you are asked for a conditional probability (e.g., 'given that the student is a boy...'), the denominator should always be the grand total.

Worked example 14 marks

80 students were asked if they study History or Geography. 21 of the 35 boys study Geography. There are 42 students in total who study Geography. Complete the two-way table and find the probability that a student chosen at random is a girl who studies History.

1
Start by drawing the table and filling in the given information: Grand Total = 80, Boys Total = 35, Boys studying Geography = 21, Geography Total = 42.
2
Calculate missing values: Girls Total = 80 - 35 = 45. History Total = 80 - 42 = 38.
3
Boys studying History = Total Boys - Boys studying Geography = 35 - 21 = 14.
4
Girls studying Geography = Total Geography - Boys studying Geography = 42 - 21 = 21.
5
Girls studying History = Total Girls - Girls studying Geography = 45 - 21 = 24. (Check: 14 + 24 = 38. Correct.)
6
The completed table looks like this:
History Geography Total
Boys 14 21 35
Girls 24 21 45
Total 38 42 80
7
Now find the probability. The number of girls who study History is 24. The grand total is 80.
8
P(Girl and History) = 24 / 80.
9
Simplify the fraction: 24/80 = 3/10 or 0.3.

	History	Geography	Total
Boys	14	21	35
Girls	24	21	45
Total	38	42	80

Recap

Two-way tables organise data for two categorical variables.
Always fill in the totals for rows and columns.
The grand total is in the bottom-right corner.
To find a probability, use the relevant number from the table over the grand total.
You can often find missing values by subtracting from the totals.

Quick check

In a two-way table of 100 people, the row totals are 40 and 60. The column totals are 30 and 70. What is the grand total?1 mark

5. Probability of Independent Events

Two events are independent if the outcome of one does not affect the outcome of the other. For example, flipping a coin and then rolling a dice are independent events. The coin landing on heads does not change the probability of the dice landing on a 6. To find the probability of two independent events BOTH happening, you multiply their individual probabilities. This is often called the 'AND' rule.

For independent events A and B: P(A and B) = P(A) × P(B)

Key term

Independent Events: Two events where the outcome of one does not affect the outcome of the other.

Examiner insight

Pay close attention to the wording 'with replacement' or 'and then puts it back'. This is the key signal from the examiner that the events are independent and you should use the multiplication rule.

Common pitfall

Adding probabilities for independent events instead of multiplying them. Remember: 'AND' means MULTIPLY, 'OR' (for mutually exclusive events) means ADD.

Worked example 12 marks

A fair coin is flipped and a fair six-sided dice is rolled. What is the probability of getting a head on the coin AND a 6 on the dice?

1
The events are independent.
2
First, find the probability of each event separately.
3
P(Head) = 1/2
4
P(6) = 1/6
5
To find the probability of both happening, multiply their probabilities.
6
P(Head and 6) = P(Head) × P(6)
7
P(Head and 6) = (1/2) × (1/6) = 1/12

Worked example 23 marks

A bag contains 5 red and 3 blue balls. A ball is picked, its colour noted, and then it is replaced. A second ball is then picked. What is the probability that both balls are red?

1
Because the first ball is replaced, the events are independent.
2
The total number of balls is 5 + 3 = 8.
3
The probability of picking a red ball on the first pick is P(Red 1) = 5/8.
4
Since the ball is replaced, the probabilities for the second pick are the same.
5
The probability of picking a red ball on the second pick is P(Red 2) = 5/8.
6
P(Red 1 and Red 2) = P(Red 1) × P(Red 2) = (5/8) × (5/8) = 25/64.

Recap

Independent events do not influence each other.
The 'AND' rule for independent events is to multiply their probabilities.
Problems involving 'with replacement' describe independent events.
Flipping coins and rolling dice are classic examples of independent events.

Quick check

The probability of winning a game is 0.2. If you play twice, what is the probability of winning both games?1 mark

6. Tree Diagrams for Combined Events

Tree diagrams are a fantastic way to visualise the outcomes of two or more events happening in sequence. Each set of branches represents an event, and the probabilities are written on the branches. To find the probability of a final outcome (a path from the start to the end), you multiply the probabilities along the branches. If you need to find the probability of more than one final outcome (e.g., the probability of getting exactly one red ball), you calculate the probability for each relevant path and then add them together.

Multiply probabilities along the branches to find the probability of a combined outcome.

Add the probabilities of the final combined outcomes to find the probability of 'this OR that'.

Key term

Tree Diagram: A diagram used to represent the probabilities of a sequence of events.

Examiner insight

Always draw the full tree diagram, even if you are only asked for one outcome. This helps you to structure your thoughts and reduces the chance of making a mistake with the probabilities on the branches.

Common pitfall

Forgetting to adjust the probabilities on the second set of branches in a 'without replacement' scenario. The denominator must decrease, as must the numerator of the item that was picked.

Worked example 14 marks

A bag contains 7 red sweets and 3 green sweets. A sweet is taken, eaten, and then a second sweet is taken. Calculate the probability that the two sweets taken are of different colours.

1
This is a 'without replacement' problem. The total number of sweets is 10.
2
Draw the first set of branches: P(Red 1) = 7/10, P(Green 1) = 3/10.
3
Draw the second set of branches. If the first was red, there are now 9 sweets left (6 red, 3 green). So, P(Red 2 | Red 1) = 6/9 and P(Green 2 | Red 1) = 3/9.
4
If the first was green, there are now 9 sweets left (7 red, 2 green). So, P(Red 2 | Green 1) = 7/9 and P(Green 2 | Green 1) = 2/9.
5
The outcomes for 'different colours' are (Red, Green) and (Green, Red).
6
Calculate the probability of each path: P(R then G) = (7/10) × (3/9) = 21/90.
7
P(G then R) = (3/10) × (7/9) = 21/90.
8
Add these probabilities together: P(different colours) = 21/90 + 21/90 = 42/90.
9
Simplify the fraction: 42/90 = 7/15.

Recap

Tree diagrams show sequential events.
Probabilities are written on the branches.
Multiply along the branches for a specific sequence (an 'AND' path).
Add the final outcome probabilities for an 'OR' condition.
For 'without replacement' problems, remember to decrease the total number (denominator) and the number of the item taken (numerator) for the second branches.

Quick check

On a tree diagram, the probabilities on a pair of branches from a single point are 0.4 and 0.6. Why?1 mark

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

A bag contains red, yellow and green counters. The probability of picking a red counter is 0.25. The probability of picking a yellow counter is three times the probability of picking a red counter. What is the probability of picking a green counter?3 marks
A biased coin is flipped 400 times. It lands on heads 250 times. What is the relative frequency of landing on tails?2 marks
The probability of a bus being on time is 0.8. Over a period of 50 days, how many times would you expect the bus to be on time?2 marks
In a group of 60 students, 40 are right-handed. There are 25 girls in the group, and 15 of them are right-handed. Complete a two-way table and find the probability that a student chosen at random is a left-handed boy.4 marks
The probability that Sam is late for school is 0.3. The probability that his bus is late is 0.4. Assuming these events are independent, what is the probability that Sam is late AND his bus is late?2 marks
A bag contains 5 red balls and 3 blue balls. A ball is taken at random, its colour is noted, and it is NOT replaced. A second ball is then taken. Draw a tree diagram and use it to find the probability that both balls are the same colour.4 marks
A spinner is split into three sections: A, B, and C. The probability of landing on A is x, on B is 2x, and on C is 0.5. Find the probability of landing on A.3 marks
A fair 8-sided dice (numbered 1-8) is rolled 240 times. How many more times would you expect to roll an odd number than you would expect to roll a number less than 3?4 marks
There are 100 members in a sports club. 60 play Tennis, 50 play Badminton, and 20 play neither. How many members play both Tennis and Badminton? Find the probability that a member chosen at random plays only Tennis.5 marks
Hannah has a 90% chance of passing her driving theory test. If she passes, she takes her practical test, which she has a 60% chance of passing. If she fails the theory, she cannot take the practical. What is the probability that Hannah gets her full driving license (passes both tests)?3 marks

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1. Mutually Exclusive Events#

2. Experimental Probability (Relative Frequency)#

3. Calculating Expected Frequency#

4. Two-Way Tables and Probability#

5. Probability of Independent Events#

6. Tree Diagrams for Combined Events#

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1. Mutually Exclusive Events

2. Experimental Probability (Relative Frequency)

3. Calculating Expected Frequency

4. Two-Way Tables and Probability

5. Probability of Independent Events

6. Tree Diagrams for Combined Events