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← Probability
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Practice deliberately

Work through Probability unit by unit, then test yourself under time.

Practice mode gives one hint and a worked explanation after every question. Test mode mixes all 7 units, holds the explanations until you submit, and then reports which units need another pass. The 8 questions printed further down this page are fixed, so you can read them and their worked answers without starting a set.

Probability coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

This page follows the Probability syllabus unit by unit. For shorter mixed retrieval outside the course sequence, use statistics practice questions with answers.

01

Probability axioms

State definitions and hypotheses precisely, construct a valid argument, and test it against a tempting counterexample.

Goal: recognise a probability axioms problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Conditional probability

Define the experiment and population, select the correct probability model, calculate with its assumptions visible, and interpret uncertainty in context.

Goal: recognise a conditional probability problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Discrete variables

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a discrete variables problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Continuous variables

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a continuous variables problem from its wording, carry out the governing method, and check that the result is reasonable.

05

Joint distributions

Define the experiment and population, select the correct probability model, calculate with its assumptions visible, and interpret uncertainty in context.

Goal: recognise a joint distributions problem from its wording, carry out the governing method, and check that the result is reasonable.

06

Expectation

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a expectation problem from its wording, carry out the governing method, and check that the result is reasonable.

07

Limit theorems

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a limit theorems problem from its wording, carry out the governing method, and check that the result is reasonable.

Probability questions with worked answers

These 8 questions are printed in full on this page and are drawn across all 7 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: Two fair six-sided dice are rolled. What is the probability that the sum is 8?

Probability · Probability axioms

  1. 5/36
  2. 1/6
  3. 1/9
  4. 6/36
Show the worked answer
  1. Rolling two dice gives 6 x 6 = 36 equally likely outcomes.
  2. The pairs that total 8 are (2,6), (3,5), (4,4), (5,3), and (6,2).
  3. That is 5 favourable outcomes.
  4. So the probability is 5/36, about 0.139.

Answer: 5/36

Question 2: A disease affects 2% of people. A test detects it 95% of the time and gives a false positive 10% of the time. Given a positive test, what is the chance the person has the disease?

Probability · Conditional probability

  1. 0.162
  2. 0.950
  3. 0.019
  4. 0.500
Show the worked answer
  1. Out of 1,000 people, about 20 have the disease and 980 do not.
  2. Of the 20, the test finds 0.95 x 20 = 19.
  3. Of the 980 healthy people, 0.10 x 980 = 98 test positive anyway.
  4. So 19 of the 19 + 98 = 117 positives are real: 19 / 117 = 0.162.

Answer: 0.162

Question 3: X follows a binomial distribution with n = 8 and p = 0.25. What is P(X = 2), to four decimal places?

Probability · Discrete variables

  1. 0.3115
  2. 0.2076
  3. 0.0625
  4. 0.1468
Show the worked answer
  1. The binomial formula uses (n choose k) times p^k times (1 - p)^(n - k).
  2. 8 choose 2 = 28, and 0.25^2 = 0.0625.
  3. 0.75^6 is about 0.177979.
  4. 28 x 0.0625 x 0.177979 = 0.3115.

Answer: 0.3115

Question 4: X is uniform on the interval from 2 to 10. What is P(X > 7)?

Probability · Continuous variables

  1. 0.375
  2. 0.300
  3. 0.700
  4. 0.625
Show the worked answer
  1. For a uniform distribution, probability is proportional to length.
  2. The full interval has length 10 - 2 = 8.
  3. The part above 7 has length 10 - 7 = 3.
  4. 3 / 8 = 0.375.

Answer: 0.375

Question 5: X and Y are independent with Var(X) = 5 and Var(Y) = 2. What is Var(2X - 3Y)?

Probability · Joint distributions

  1. 38
  2. 4
  3. 28
  4. 16
Show the worked answer
  1. Constants come out of a variance squared, and independence means no covariance term.
  2. Var(2X) = 2^2 x 5 = 20.
  3. Var(-3Y) = (-3)^2 x 2 = 18. Squaring removes the minus sign.
  4. 20 + 18 = 38.

Answer: 38

Question 6: Each attempt succeeds with probability 0.2, independently. What is the expected number of attempts until the first success?

Probability · Expectation

  1. 5
  2. 0.2
  3. 4
  4. 10
Show the worked answer
  1. This is a geometric random variable counting trials up to and including the first success.
  2. Its expected value is 1 / p.
  3. 1 / 0.2 = 5.
  4. So on average it takes 5 attempts.

Answer: 5

Question 7: A population has mean 50 and standard deviation 10. For a sample of 100, what is the approximate probability the sample mean exceeds 52?

Probability · Limit theorems

  1. 0.0228
  2. 0.4207
  3. 0.1587
  4. 0.0013
Show the worked answer
  1. The central limit theorem makes the sample mean roughly normal.
  2. Its standard deviation is 10 divided by the square root of 100, which is 1.
  3. The z score is (52 - 50) / 1 = 2.
  4. The area above z = 2 in a standard normal table is 0.0228.

Answer: 0.0228

Question 8: A waiting time is exponential with mean 4 minutes. What is P(X > 6), to four decimal places?

Probability · Continuous variables

  1. 0.2231
  2. 0.7769
  3. 0.6667
  4. 0.0498
Show the worked answer
  1. An exponential with mean 4 has rate 1/4.
  2. The survival probability is e raised to the power of -(rate x time).
  3. Rate times time is 6 / 4 = 1.5.
  4. e^(-1.5) = 0.2231.

Answer: 0.2231

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed Probability questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1Probability axiomsTwo fair six-sided dice are rolled. What is the probability that the sum is 8?5/36
2Conditional probabilityA disease affects 2% of people. A test detects it 95% of the time and gives a false positive 10% of the time. Given a positive test, what is the chance the person has the disease?0.162
3Discrete variablesX follows a binomial distribution with n = 8 and p = 0.25. What is P(X = 2), to four decimal places?0.3115
4Continuous variablesX is uniform on the interval from 2 to 10. What is P(X > 7)?0.375
5Joint distributionsX and Y are independent with Var(X) = 5 and Var(Y) = 2. What is Var(2X - 3Y)?38
6ExpectationEach attempt succeeds with probability 0.2, independently. What is the expected number of attempts until the first success?5
7Limit theoremsA population has mean 50 and standard deviation 10. For a sample of 100, what is the approximate probability the sample mean exceeds 52?0.0228
8Continuous variablesA waiting time is exponential with mean 4 minutes. What is P(X > 6), to four decimal places?0.2231

How the generated Probability sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this Probability practice page

What does this page cover?

It covers all 7 Probability units listed above. Choose one unit for focused work, or mixed review to test method selection, and switch to test mode when you want all units mixed under time.

When can I see correct answers and explanations?

The 8 printed questions on this page keep their worked answers behind a toggle, so you can check any one of them straight away. In the generator above, practice mode explains each question as soon as you answer it, while test mode holds every explanation until you submit.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.