Learn Probability unit by unit

Model uncertainty with random variables and distributions.

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Units at a glance

Every unit in this course, what you will be able to do once you finish it, and roughly when it lands in a 14-week schedule.

Probability course map: 7 units
#	Unit	What you can do after it	Weeks
01	Probability axioms	You can lay out a sample space, assign probabilities that obey the basic rules, and compute the chance of an event.	Weeks 1-2
02	Conditional probability	You can update a chance once you learn something new, and run Bayes' rule.	Weeks 3-4
03	Discrete variables	You can use binomial, geometric and Poisson models for counts, and find their means and variances.	Weeks 5-6
04	Continuous variables	You can use density curves where probability is area under a curve, including uniform, exponential and normal shapes.	Weeks 7-8
05	Joint distributions	You can handle two random variables at once, pull out each one's own distribution, and test independence.	Weeks 9-10
06	Expectation	You can find means and variances of sums and rescaled variables without rebuilding the whole distribution.	Weeks 11-12
07	Limit theorems	You can explain why sample averages cluster near the true mean and turn bell-shaped.	Weeks 13-14

Probability course units

Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.

Probability axioms

Weeks 1-2

You can lay out a sample space, assign probabilities that obey the basic rules, and compute the chance of an event. You can also use complements and unions correctly.

List the sample space for two dice and count the outcomes.
Use the fact that the chance of not-A is 1 minus the chance of A.
Apply inclusion-exclusion to two overlapping events.

Worked example

Two fair dice are rolled. What is the chance the total is 7?

There are 36 equally likely ordered outcomes
Totals of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — six of them

Answer6/36 = 1/6

Most common mistakeCounting unordered pairs only: finding 3 pairs out of 21 gives about 0.143, because it treats (2,5) and (5,2) as the same single outcome.

Explain probability axioms Practice probability axioms

Conditional probability

Weeks 3-4

You can update a chance once you learn something new, and run Bayes' rule. You can also test whether two events really are independent.

Compute the chance of A given B as the joint chance divided by the chance of B.
Apply Bayes' rule to a medical test with a rare condition.
Check independence by comparing the joint chance to the product.

Worked example

P(A) = 0.4, P(B) = 0.5, P(A and B) = 0.2. Find P(A given B).

Divide the joint probability by P(B): 0.2 / 0.5
= 0.4

Answer0.4, which equals P(A), so A and B are independent

Most common mistakeDividing by the wrong event: computing 0.2/0.4 = 0.5, which is the chance of B given A, not A given B.

Explain conditional probability Practice conditional probability

Discrete variables

Weeks 5-6

You can use binomial, geometric and Poisson models for counts, and find their means and variances. You can also choose the right model from the story alone.

Spot a Poisson setting: rare events over a fixed stretch of time.
Compute a binomial probability with the combination formula.
Find the mean of a geometric variable as 1 divided by the success chance.

Worked example

Flip a fair coin 5 times. Find the chance of exactly 3 heads.

Number of ways to pick which 3 flips are heads: 10
Each specific pattern has chance 0.5 to the fifth = 0.03125

Answer10 x 0.03125 = 0.3125

Most common mistakeLeaving out the count of arrangements: reporting 0.03125, which is the chance of one exact order such as heads-heads-heads-tails-tails.

Explain discrete variables Practice discrete variables

Continuous variables

Weeks 7-8

You can use density curves where probability is area under a curve, including uniform, exponential and normal shapes. You can also find percentiles with z-scores.

Find the chance an exponential variable exceeds a given value.
Convert a value into a z-score and look up its area.
Explain why the chance of hitting one exact value is zero.

Worked example

X is normal with mean 70 and standard deviation 8. Find the chance X is under 78.

z = (78 - 70)/8
z = 1, and the area left of z = 1 is 0.8413

AnswerAbout 84%

Most common mistakeSubtracting without dividing by the standard deviation: using z = 8 puts the answer at essentially 100%, when the true figure is 84%.

Explain continuous variables Practice continuous variables

Joint distributions

Weeks 9-10

You can handle two random variables at once, pull out each one's own distribution, and test independence. You can also compute a covariance.

Sum a joint table across a row to get a marginal distribution.
Compare each joint entry with the product of its marginals.
Compute covariance as E[XY] minus E[X] times E[Y].

Worked example

A joint table has P(X=1, Y=1) = 0.3, with P(X=1) = 0.5 and P(Y=1) = 0.6. Is this cell consistent with independence?

If independent, the joint would be 0.5 x 0.6
0.5 x 0.6 = 0.30, which matches

AnswerThis cell is consistent, but every other cell has to match too

Most common mistakeDeclaring independence from one matching cell: independence requires every cell to equal the product of its marginals, and one match proves nothing.

Explain joint distributions Practice joint distributions

Expectation

Weeks 11-12

You can find means and variances of sums and rescaled variables without rebuilding the whole distribution. You can also add expectations even when the variables are linked.

Use E[aX + b] = a E[X] + b.
Use Var(aX + b) = a squared times Var(X).
Add expectations of dependent variables using linearity.

Worked example

E[X] = 10 and Var(X) = 4. Find E[3X + 5] and Var(3X + 5).

Mean: 3(10) + 5 = 35
Variance: 3 squared times 4 = 36

AnswerMean 35, variance 36

Most common mistakeAdding the 5 to the variance as well: reporting 3(4) + 5 = 17, when sliding a variable up by 5 does not change how spread out it is.

Explain expectation Practice expectation

Limit theorems

Weeks 13-14

You can explain why sample averages cluster near the true mean and turn bell-shaped. You can also use that bell shape to approximate a count.

Apply the central limit theorem to a sample mean from 100 observations.
Compute the standard error as sigma over the square root of n.
Use a normal approximation to a binomial when both np and n(1-p) are large.

Worked example

The population standard deviation is 20 and the sample size is 100. Find the standard error of the mean.

Standard error = sigma / square root of n
20 / 10

Answer2

Most common mistakeDividing by n rather than the square root of n: reporting 0.2, ten times too small, which makes every interval look impossibly precise.

Explain limit theorems Practice limit theorems

Prepare for Probability practice

Start with the earliest uncertain prerequisite

Before timing yourself, check whether you can explain Probability axioms from a blank page. Then connect it to Conditional probability. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.

Use tools to verify, not to choose the method for you

The Statistics calculator can test calculations and representations used in Probability. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.

Know when to move to the full test

Move from Probability practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.

Before and after the syllabus

Learn the ideas, then practise them

The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.

GuidesStatistics explainedConcepts, worked methods and checks behind this course.

PracticeMixed Statistics questionsAnswer review after every attempt.

ReferenceFormulas for this courseConditions and a worked example for each one.

Assess

Take the complete Probability test

Begin test →

Reference

Review essential formulas

Open library →

Calculate

Use the Statistics calculator

Open tool →

Plan

Prepare around your exam date

Build plan →

Questions about the Probability course

Where should I start?

Start with Probability axioms if you are following the full sequence. If that unit feels automatic, open the Probability practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.

How do I know I am ready for the course test?

You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Probability test only after you can correct practice errors from a blank page.

Which calculator supports this course?

The Statistics calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.

Learn Probability unit by unit

Units at a glance

Probability course units

Probability axioms

Worked example

Conditional probability

Worked example

Discrete variables

Worked example

Continuous variables

Worked example

Joint distributions

Worked example

Expectation

Worked example

Limit theorems

Worked example

Prepare for Probability practice

Start with the earliest uncertain prerequisite

Use tools to verify, not to choose the method for you

Know when to move to the full test

Learn the ideas, then practise them

Other university courses

Mathematical Statistics

Numerical Methods

Real Analysis

Take the complete Probability test

Review essential formulas

Use the Statistics calculator

Prepare around your exam date

Questions about the Probability course