Math · Unit 15

Probability and Statistics

Probability, counting, distributions and sampling

Probability is the mathematics of uncertainty, and it is famously counter-intuitive: trained intuition gets it wrong often enough that the subject exists partly to correct it.

The unit covers probability basics, counting and combinations, data distributions, spread and standard deviation, and sampling and inference.

This unit breaks down into 21 short steps and 120 questions, starting at difficulty 3 and building to 4. Below you can see exactly what it covers, how the path is structured, and worked examples with explanations.

Steps: 21
Questions: 120
Difficulty: 3-4

What this unit covers

Spread and Standard Deviation
Sampling and Inference
Data Distributions
Probability Basics
Counting and Combinations

Where this fits

Needs Ratio, Proportion and Percent. Underpins Science's Research Methods and Statistics.

Where people slip

Independent events have no memory. A coin that has landed heads nine times is exactly as likely to land heads again, and expecting otherwise is the gambler's fallacy.

How the unit is structured

Probability and Statistics runs as 21 short steps that unlock in order. 15 are practice rounds and 6 are challenge rounds that pull together everything before them. Questions start at difficulty 3 and climb to 4 as you progress.

Step 1 · easierStep 21 · harder

Challenge rounds

Example questions

30 real questions from this unit, with the answer and the reason behind it, grouped by what they practise. There are 120 in the unit altogether.

Counting and Combinations

Fact or fibLevel 3
1. 4! (4 factorial) equals 24.
Answer: True
4! = 4 x 3 x 2 x 1 = 24.
Fill the blankLevel 3
2. 3! = 3 x 2 x 1 = ____.
- 6correct
- 9
- 3
- 5
3 x 2 x 1 = 6.
Guess the numberLevel 3
3. How many different ways can 4 different runners finish in 1st, 2nd, 3rd, and 4th place?
Answer: 24
There are 4! = 4 x 3 x 2 x 1 = 24 possible finishing orders.
Match the pairsLevel 3
4. Match each factorial with its value.
Answer: 3! = 6; 4! = 24; 5! = 120
3! = 6, 4! = 24, and 5! = 120.
Multiple choiceLevel 3
5. How many ways can you choose 2 toppings from 4 different toppings when order does not matter?
- 6correct
- 8
- 12
- 4
The number of combinations is C(4,2) = (4 x 3)/(2 x 1) = 6.
Odd one outLevel 3
6. Which of these does NOT equal 6?
- 4!correct
- 3!
- 3 x 2 x 1
- 2 x 3
4! = 24, while 3!, 3 x 2 x 1, and 2 x 3 all equal 6.

Data Distributions

Build the sentenceLevel 3
7. Build a true sentence about the mode.
Answer: The mode is the most common value
The mode is the value that appears most often.
Fact or fibLevel 3
8. The mode is the value that appears most often in a data set.
Answer: True
Right, the mode is the most common value.
Fill the blankLevel 3
9. The ____ of a data set is the sum of all values divided by the number of values.
- meancorrect
- mode
- median
- range
The mean is the average, found by dividing the total by the count.
Match the pairsLevel 3
10. For the data set 2, 2, 3, 5, 8, match each measure to its value.
Answer: Mean = 4; Median = 3; Mode = 2
The mean is 20/5 = 4, the middle value is 3, and 2 appears most often.
Multiple choiceLevel 3
11. Which value appears most often in a data set?
- Modecorrect
- Mean
- Median
- Range
The mode is the value that occurs most frequently.
Choose all that applyLevel 4
12. Which of these are measures of center? Select all that apply.
- Meancorrect
- Mediancorrect
- Modecorrect
- Range
Mean, median, and mode describe the center; range measures spread.

Probability Basics

Fill the blankLevel 3
13. The probability of an event that is certain to happen equals ____.
- 1correct
- 0
- 1/2
- 100
A certain event has a probability of 1.
Match the pairsLevel 3
14. Match each event with its probability on fair equipment.
Answer: Heads on a coin = 1/2; Rolling a 3 on a die = 1/6; Rolling any number 1 to 6 = 1
A coin gives 1/2, one die face is 1/6, and a sure outcome is 1.
Multiple choiceLevel 3
15. A standard six-sided die is rolled once. What is the probability of rolling a 3?
- 1/6correct
- 1/3
- 3/6
- 1/2
There is 1 way to roll a 3 out of 6 equally likely outcomes, so the probability is 1/6.
Odd one outLevel 3
16. Which of these numbers could NOT be a probability?
- 1.2correct
- 0
- 0.45
- 1
Probabilities range from 0 to 1, so 1.2 is impossible.
Build the sentenceLevel 4
17. Build a true sentence about independent events.
Answer: Independent events do not affect each other
With independent events, one outcome does not change the other's probability.
Choose all that applyLevel 4
18. For a fair six-sided die, which of these statements are true? Select all that apply.
- P(rolling a 6) = 1/6correct
- P(rolling an odd number) = 1/2correct
- P(rolling a number less than 7) = 1correct
- P(rolling a 7) = 1/6
A 7 is impossible on a normal die, so its probability is 0, not 1/6.

Sampling and Inference

Build the sentenceLevel 3
19. Build a true sentence about samples.
Answer: A sample is part of a population
A sample is a subset selected from the population.
Choose all that applyLevel 3
20. Which of these describe a good sample? Select all that apply.
- It should be chosen randomlycorrect
- It should represent the populationcorrect
- It must include the whole population
- More bias makes it better
Good samples are chosen randomly and represent the population; a sample is not the whole population, and less bias is better.
Fact or fibLevel 3
21. A statistic describes a whole population, not a sample.
Answer: False
A statistic describes a sample; a parameter describes the whole population.
Fill the blankLevel 3
22. A number that describes a whole population is called a ____.
- parametercorrect
- statistic
- sample
- outcome
A parameter summarizes a population; a statistic summarizes a sample.
Multiple choiceLevel 3
23. The complete group you want to study is called the...?
- Populationcorrect
- Sample
- Statistic
- Outcome
The population is the entire group of interest; a sample is just part of it.
Odd one outLevel 4
24. Which of these is NOT a random sampling method?
- Convenience samplingcorrect
- Simple random sampling
- Stratified sampling
- Systematic sampling
Convenience sampling picks whoever is easy to reach, so it is not random.

Spread and Standard Deviation

Build the sentenceLevel 3
25. Build a true sentence about the range.
Answer: The range is the highest value minus the lowest
The range is the highest value minus the lowest value.
Choose all that applyLevel 3
26. Which of these are measures of how spread out data is? Select all that apply.
- Rangecorrect
- Standard deviationcorrect
- Mean
- Mode
Range and standard deviation measure spread; mean and mode describe the center.
Fact or fibLevel 3
27. If every value in a data set is the same, the standard deviation is 0.
Answer: True
With no variation between values, the spread is 0.
Fill the blankLevel 3
28. The ____ is the difference between the largest and smallest values in a data set.
- rangecorrect
- mean
- mode
- median
Range = maximum - minimum.
Multiple choiceLevel 3
29. What is the range of the data set 4, 9, 15, 20?
- 16correct
- 11
- 20
- 4
The range is the largest value minus the smallest: 20 - 4 = 16.
Guess the numberLevel 4
30. Find the range of this data set: 12, 19, 25, 40, 47.
Answer: 35
Range = 47 - 12 = 35.

Where these questions come from. Each unit starts as a plan of the concepts it should cover and the difficulty it should span. Questions are written against that plan with AI assistance, then checked by a validator that rejects anything without a single defensible answer, an explanation, or plausible wrong options. How we write questions sets out the whole process, and corrections are fixed in the bank and reach the site and the app the same day.

How you practise

This unit mixes 14 different question formats, so you are recalling and applying rather than recognising the same layout every time.

Build the sentence
Choose all that apply
Fact or fib
Fill the blank
Guess the number
Match the pairs
Multiple choice
Odd one out
Picture question
Put in order
Sort into groups
Spell it
True or false
Type the answer

Practise Probability and Statistics

120 questions across 21 steps. Start with step one and crawl at your own pace.

Play this unit

Read about Probability and Statistics

Explainers from our blog on what this unit covers. Each one ends with real questions from the bank.

More units in Math

See all 18 units in Math →