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.
01
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.
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.
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%.
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.
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.
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.
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.
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.
