Practice test — Probability.
20 questions drawn at random from the whole Probability programme, mixed as in a real exam sitting: scoring +1/−0.25/0. At the end you receive by email the correct answer to every question and your weak points by topic — with an account you see them straight away on this page (the guided solution unlocks with your first lesson).
Each test mixes every topic of one first-year course at an Italian university, like a real exam sitting.
Want to see the reasoning before you try? Look at the interactive probability tree.
Test in ItalianThe questions of this simulator are in Italian, as in the real exam.
Topics covered by the test
- Axioms and calculating probabilities
- Combinatorics
- Conditional probability and Bayes’ theorem
- Discrete and continuous random variables
- Expected value and variance
- Law of large numbers and central limit theorem
- Markov chains (basics)
Two worked examples from the test
Per gli assiomi di Kolmogorov, la probabilità di un evento soddisfa sempre:
Solution
Il primo assioma di Kolmogorov impone che ogni probabilità sia un numero reale compreso tra 0 e 1 inclusi, qualunque sia l'evento considerato.
Lanciando un dado equo a 6 facce, è:
Solution
Pari: . Maggiore di 4: . Unione: , 4 esiti su 6, quindi Attenzione a non sommare semplicemente senza sottrarre l'intersezione contata due volte.
In the online practice test you will find 18 more questions like these; the correct answer to every question arrives by email.
What the Probability exam is like
The written paper alternates a combinatorics exercise, a conditional-probability problem solved with Bayes (often the classic diagnostic test with false positives), the expected value and variance of a random variable, and sometimes a question on a simple Markov chain (transition matrix, stationary distribution). The oral insists on proofs of the basic properties (linearity of expectation, the Cauchy–Schwarz inequality).
Where the exam is lost
Confusing independence with mutual exclusivity: they are the opposite — independent events typically overlap, mutually exclusive events never overlap but are rarely independent. In combinatorics, the classic mistake is choosing arrangements instead of combinations (or vice versa) without asking whether order matters. In Bayes, the trap is updating the probability on the evidence while forgetting the law of total probability in the denominator.
How to read your score, in short.
What are the questions of the Probability test like?
20 multiple-choice questions (4 options), drawn at random from a larger bank and spread over the WHOLE course programme together, not one topic at a time: axioms and calculating probabilities, combinatorics, conditional probability and Bayes’ theorem, discrete and continuous random variables, expected value and variance, law of large numbers and central limit theorem, Markov chains (basics).
Is there a penalty for wrong answers?
Yes: +1 point for a correct answer, −0.25 for a wrong one, 0 for a question left blank — as in the TOLC (the Italian university admission tests). You should answer only when you are reasonably sure: it is the same logic as a real multiple-choice exam.
What does the final score mean?
15 out of 20 or more: you are ready, the programme is solid. 10-14: needs consolidating — review the topics of the questions you got wrong and try again. Under 10: to be reviewed calmly before you sit the test again.
Course still weak?
Let us look at your results together — the diagnostic is free — and I will tell you honestly what it takes to close the gap.