Probability for interviews quiz

Question 1

What is the difference between independence and mutual exclusivity of two events A and B?

Accepted Answer

Independence: P(A∩B)=P(A)P(B); mutual exclusivity: P(A∩B)=0, and they are generally incompatible for positive-probability events

Answer

They are the same concept stated two ways

Answer

Mutually exclusive events are always independent

Answer

Independent events always have P(A∩B)=0

Question 2

State Bayes' theorem and explain the role of the prior, likelihood, and evidence.

Accepted Answer

P(H|D)=P(D|H)P(H)/P(D)

Answer

P(H|D)=P(D|H)+P(H)

Answer

P(H|D)=P(H)/P(D|H)

Answer

P(H|D)=P(D)P(H)/P(D|H)

Question 3

A disease affects 1% of a population. A test has 99% sensitivity and 95% specificity. Given a positive test, what is the probability the person has the disease?

Accepted Answer

About 16.7%

Answer

About 99%

Answer

About 50%

Answer

About 1%

Question 4

What is the expected number of coin flips of a fair coin to get the first head?

Accepted Answer

2

Answer

1

Answer

4

Answer

0.5

Question 5

Explain the difference between a probability density function (PDF) and a probability mass function (PMF).

Accepted Answer

PMF is for discrete vars (gives P(X=x)); PDF is for continuous vars (density, integrate to get probability)

Answer

PDF gives exact point probabilities; PMF gives densities

Answer

Both give point probabilities that must be ≤1

Answer

A PDF must always be ≤1 at every point

Question 6

What is the Law of Total Expectation (tower property)?

Accepted Answer

E[X]=E[E[X|Y]]

Answer

E[X]=E[X|Y]

Answer

E[X]=Var(E[X|Y])

Answer

E[X]=E[X]E[Y]

Question 7

State the Central Limit Theorem and one key condition for it to hold.

Accepted Answer

Requires i.i.d. samples with finite mean and finite variance

Answer

Requires the population to be normal

Answer

Requires only finite mean, variance can be infinite

Answer

Holds for any distribution including Cauchy

Question 8

What is the variance of a Bernoulli(p) random variable, and at what p is it maximized?

Accepted Answer

Var=p(1-p), maximized at p=0.5

Answer

Var=p, maximized at p=1

Answer

Var=p²(1-p), maximized at p=2/3

Answer

Var=1-p, maximized at p=0

Question 9

You roll two fair dice. What is the probability the sum is 7?

Accepted Answer

1/6

Answer

1/12

Answer

1/8

Answer

1/9

Question 10

Explain the difference between covariance and correlation.

Accepted Answer

Correlation is covariance scaled by the product of standard deviations, bounded in [-1,1]

Answer

They are identical

Answer

Covariance is always between -1 and 1

Answer

Correlation has the units of X times Y

Question 11

What does it mean for two random variables to be conditionally independent given a third?

Accepted Answer

P(X,Y|Z)=P(X|Z)P(Y|Z); knowing Z removes any dependence between X and Y

Answer

It is the same as marginal independence

Answer

It always implies X and Y are marginally independent

Answer

It means P(X,Y)=P(X)P(Y)

Question 12

In the Monty Hall problem, should you switch doors after the host reveals a goat, and what is the win probability?

Accepted Answer

Switch; win probability becomes 2/3

Answer

Stay; both are 1/2

Answer

Switch; win probability becomes 1/2

Answer

It does not matter; always 1/3

Question 13

What is the memoryless property, and which distributions have it?

Accepted Answer

Exponential (continuous) and geometric (discrete)

Answer

Normal (continuous) and Poisson (discrete)

Answer

Uniform (continuous) and binomial (discrete)

Answer

Gamma (continuous) and negative binomial (discrete)

Question 14

Given a Poisson process with rate λ events per unit time, what is the distribution of inter-arrival times?

Accepted Answer

Exponential(λ) with mean 1/λ

Answer

Poisson(λ)

Answer

Uniform on [0,1/λ]

Answer

Normal(λ, λ)

Question 15

What is the expected value and variance of a continuous Uniform(a,b) distribution?

Accepted Answer

Mean (a+b)/2, Var (b-a)²/12

Answer

Mean (a+b)/2, Var (b-a)/2

Answer

Mean (b-a)/2, Var (b-a)²/6

Answer

Mean ab, Var (b-a)²/4

Question 16

Explain the difference between MLE and MAP estimation from a probabilistic standpoint.

Accepted Answer

MLE maximizes the likelihood; MAP maximizes the posterior (likelihood × prior)

Answer

MLE uses a prior; MAP does not

Answer

MAP maximizes P(D|θ); MLE maximizes P(θ|D)

Answer

They always give identical results

Question 17

What is the probability that in a group of 23 people at least two share a birthday (birthday paradox)?

Accepted Answer

About 50%

Answer

About 6%

Answer

About 23%

Answer

About 99%

Question 18

What is a confidence interval, and is it correct to say there is a 95% probability the parameter lies in a particular 95% CI?

Accepted Answer

No; 95% refers to the long-run coverage of the procedure, not a probability about a fixed parameter in one interval

Answer

Yes, that statement is exactly the frequentist definition

Answer

A CI is the same as a Bayesian credible interval

Answer

The parameter is random, so the statement is always true

Question 19

Two players alternately flip a fair coin; the first to get heads wins. What is the probability the first player wins?

Accepted Answer

2/3

Answer

1/2

Answer

3/4

Answer

1/3

Question 20

What is the difference between a Type I and Type II error?

Accepted Answer

Type I: false positive (reject true null, α); Type II: false negative (fail to reject false null, β)

Answer

Type I: false negative (β); Type II: false positive (α)

Answer

Both refer to the same error at different thresholds

Answer

Type I error probability is power

Question 21

If X ~ N(0,1), what is the distribution of X², and what is its mean?

Accepted Answer

Chi-squared with 1 df, mean 1

Answer

Normal with mean 0

Answer

Exponential with mean 1

Answer

Gamma(2,1), mean 2

Question 22

Explain the difference between sampling with and without replacement in terms of the resulting distributions.

Accepted Answer

With replacement → binomial (independent); without → hypergeometric (dependent)

Answer

Both yield binomial counts

Answer

With replacement → hypergeometric; without → binomial

Answer

Both yield Poisson counts

Question 23

What is the expected number of draws to collect all n coupons in the coupon collector problem?

Accepted Answer

n·H_n ≈ n ln n

Answer

n

Answer

n²

Answer

2^n

Question 24

What is the difference between a marginal, joint, and conditional distribution?

Accepted Answer

Joint = P(X,Y); marginal = summing out a variable; conditional = P(X,Y)/P(Y)

Answer

Marginal = P(X|Y); conditional = P(X)

Answer

They are three names for the same distribution

Answer

Joint = P(X)+P(Y)

Question 25

For independent X and Y, how do means and variances of X+Y combine, and what about X-Y?

Accepted Answer

For independent vars, Var(X+Y)=Var(X-Y)=Var(X)+Var(Y)

Answer

Var(X-Y)=Var(X)-Var(Y) for independent vars

Answer

Var(X+Y)=Var(X)+Var(Y) but Var(X-Y)=0

Answer

Means subtract but variances always cancel

Question 26

What is the moment generating function (MGF) and why is it useful?

Accepted Answer

M_X(t)=E[e^{tX}]; derivatives at 0 give moments and it uniquely identifies the distribution

Answer

M_X(t)=E[tX], gives the mean only

Answer

M_X(t)=Var(X)·t

Answer

It is the same as the cumulative distribution function

Question 27

You have a biased coin with unknown bias. How can you simulate a fair coin using von Neumann's trick?

Accepted Answer

Flip twice; HT→heads, TH→tails, discard HH/TT and repeat

Answer

Flip once and take the result

Answer

Flip three times and take the majority

Answer

Add a constant 0.5 to each flip

Question 28

What is Simpson's paradox, and why does it matter for data analysis?

Accepted Answer

A subgroup trend that reverses when data is aggregated, caused by a confounder

Answer

A trend that is identical at every level of aggregation

Answer

A property of normal distributions only

Answer

A type of sampling without replacement

Question 29

What is the relationship between the Poisson and Binomial distributions?

Accepted Answer

Poisson is the limit of Binomial(n,p) as n→∞, p→0, np=λ fixed

Answer

Poisson is the limit of Binomial as p→1

Answer

Binomial is the limit of Poisson as λ→0

Answer

They are unrelated

Question 30

Define entropy of a discrete random variable and state when it is maximized.

Accepted Answer

H(X)=-Σ p(x) log p(x); maximized by the uniform distribution (H=log k)

Answer

H(X)=Σ p(x); maximized when deterministic

Answer

H(X)=Var(X); maximized for skewed distributions

Answer

Entropy is always 1 bit

Test yourself