Self-Assessment Quiz: Discrete Probability

Twenty questions to check your understanding. Answer before opening the key. Aim for 16+.

Question 1

The sample space of an experiment is:

A) the most likely outcome B) the set of all mutually-exclusive, exhaustive outcomes C) any subset of outcomes you care about D) the probability assigned to each outcome

Question 2

An event is formally defined as:

A) a single outcome B) a probability between 0 and 1 C) a subset of the sample space D) a random variable

Question 3

The equally-likely (Laplace) formula $P(E) = \lvert E \rvert / \lvert S \rvert$ is valid:

A) for every sample space B) only when the outcomes of $S$ are equally likely C) only when $E$ is a single outcome D) only for infinite sample spaces

Question 4

By the complement rule, $P(\overline{E})$ equals:

A) $P(E)$ B) $1 - P(E)$ C) $1/P(E)$ D) $P(E) - 1$

Question 5

For events $E$ and $F$, the general formula $P(E \cup F) = P(E) + P(F) - P(E \cap F)$ is needed instead of $P(E) + P(F)$ precisely when:

A) $E$ and $F$ are disjoint B) $E$ and $F$ overlap (are not disjoint) C) $E \subseteq F$ D) the outcomes are not equally likely

Question 6

A random variable $X$ on a sample space $S$ is:

A) a randomly chosen number B) a subset of $S$ C) a function $X \colon S \to \mathbb{R}$ D) the probability of an event

Question 7

For an indicator random variable $\mathbf{1}_A$, the value $E[\mathbf{1}_A]$ equals:

A) $1$ B) $0$ C) $P(A)$ D) $\lvert A \rvert$

Question 8

Linearity of expectation, $E[X + Y] = E[X] + E[Y]$, requires:

A) $X$ and $Y$ to be independent B) $X$ and $Y$ to be indicators C) no special hypothesis — it always holds D) $X$ and $Y$ to have the same distribution

Question 9

In the hat-check problem ($n$ people, random hat returns), the expected number of people who get their own hat back is:

A) $0$ B) $1$ C) $n/2$ D) $n$

Question 10

The expected number of empty buckets when $n$ keys are hashed uniformly into $m$ buckets is:

A) $m/n$ B) $m\left(1 - \tfrac{1}{m}\right)^n$ C) $n\left(1 - \tfrac{1}{n}\right)^m$ D) $m - n$

Question 11

The computational formula for variance is:

A) $\operatorname{Var}(X) = E[X] - (E[X])^2$ B) $\operatorname{Var}(X) = (E[X])^2 - E[X^2]$ C) $\operatorname{Var}(X) = E[X^2] - (E[X])^2$ D) $\operatorname{Var}(X) = E[(X - \mu)]^2$

Question 12

Two random variables both have $E[X] = 100$, but one is constant and one swings between 0 and 200. What distinguishes them?

A) their expectation B) their variance / standard deviation C) their sample space size D) nothing — they are identical

Question 13

Variance can never be negative because:

A) probabilities are at most 1 B) it is the expectation of a squared (hence non-negative) quantity C) the mean is always positive D) standard deviation is its square root

Question 14

The probabilistic method concludes that an object with property $Q$ exists by showing:

A) $P(\text{random object has } Q) = 1$ B) $P(\text{random object has } Q) > 0$ C) every random object has $Q$ D) $E[Q] = 0$

Question 15

The probabilistic method is, at its logical core, an instance of:

A) proof by induction B) proof by contradiction C) direct proof D) a counterexample

Question 16 (True/False, justify)

True or false: If you run a Monte-Carlo simulation 10 million times and the empirical probability comes out to $0.167$, you have proved that the true probability is exactly $1/6$. Justify in one sentence.

Question 17 (True/False, justify)

True or false: $E[X^2] = (E[X])^2$ for every random variable $X$. Justify in one sentence (and name what the difference between the two sides measures).

Question 18 (True/False, justify)

True or false: For the experiment of flipping two fair coins, the correct equally-likely sample space has three outcomes: zero heads, one head, two heads. Justify.

Question 19 (Short answer)

State the three probability axioms (non-negativity, normalization, additivity) in your own words.

Question 20 (Short answer)

In two or three sentences, explain the general strategy "write it as a sum of indicators, then add up the probabilities," and why it is so powerful for computing expected values.

Answer Key

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