Basic Probability Self-Test

Probability Theory Quiz

Test your knowledge of probability concepts

Q1

If $P(A) = 0.5$, $P(B) = 0.4$, and $P(A \cap B) = 0.2$, what is $P(A|B)$?

0.3 0.5 0.4 0.6

Q2

According to the multiplication theorem, $P(A \cap B) = ?$

$P(A) + P(B)$ $P(A) \times P(B|A)$ $P(A) - P(B)$ $P(A) \div P(B)$

Q3

Two events A and B are independent if:

$P(A \cap B) = P(A) + P(B)$ $P(A \cap B) = P(A) \times P(B)$ $P(A|B) = P(B|A)$ $P(A) = P(B)$

Q4

Bayes' Theorem relates:

Conditional and marginal probabilities Only independent events Mutually exclusive events Only joint probabilities

Q5

The theorem of total probability requires that the events $B_1, B_2, ..., B_n$ are:

Independent Mutually exclusive and exhaustive Dependent Complementary

Q6

A random variable is:

A fixed number A variable that takes numerical values based on random outcomes Always discrete A constant in probability

Q7

The mean (expected value) of a discrete random variable X is calculated as:

$\sum x^2P(x)$ $\sum xP(x)$ $\sum P(x)$ $\sum (x-\mu)^2P(x)$

Q8

Variance measures:

Central tendency Spread or dispersion Skewness Kurtosis

Q9

Standard deviation is:

Square of variance Square root of variance Twice the variance Half the variance

Q10

Bernoulli trials are characterized by:

Three possible outcomes Two possible outcomes (success/failure) Continuous outcomes Dependent trials

Q11

The binomial distribution applies to:

Continuous random variables Fixed number of independent Bernoulli trials Dependent trials Infinite trials

Q12

The formula for conditional probability $P(A|B)$ is:

$\frac{P(A \cap B)}{P(B)}$ $\frac{P(A)}{P(B)}$ $\frac{P(B)}{P(A)}$ $\frac{P(A \cup B)}{P(B)}$

Q13

If $P(A) = 0.3$, $P(B) = 0.4$, and $P(A \cap B) = 0.12$, are A and B independent?

Yes No Cannot determine Only if $P(A \cup B) = 1$

Q14

Bayes' formula is $P(A|B) = ?$

$\frac{P(B|A)P(A)}{P(B)}$ $\frac{P(A)P(B)}{P(A \cap B)}$ $\frac{P(A \cap B)}{P(A)}$ $\frac{P(B|A)}{P(A)}$

Q15

If $P(B_1) = 0.6$, $P(B_2) = 0.4$, $P(A|B_1) = 0.3$, $P(A|B_2) = 0.5$, then $P(A) = ?$

0.38 0.42 0.35 0.40

Q16

Which of the following is a discrete random variable?

Height of a person Weight of a package Number of heads in 10 coin tosses Time taken to run a race

Q17

If X has values 1, 2, 3 with probabilities 0.2, 0.5, 0.3 respectively, $E(X) = ?$

2.1 2.0 1.9 2.2

Q18

For a discrete random variable, $\text{Var}(X) = ?$

$E[X^2] - (E[X])^2$ $E[X] - E[X^2]$ $E[X^2] + (E[X])^2$ $(E[X])^2 - E[X^2]$

Q19

Standard deviation has the same units as:

The random variable The square of the random variable Probability Variance

Q20

The binomial distribution has parameters:

$n$ and $\mu$ $n$ and $p$ $\mu$ and $\sigma$ $p$ and $q$

Q21

For a binomial distribution with $n=5$, $p=0.4$, $P(X=3) = ?$

0.2304 0.3456 0.2592 0.1536

Q22

If $P(A) = 0.6$, $P(B) = 0.5$, and A and B are independent, $P(A \cap B) = ?$

0.3 0.5 0.6 1.1

Q23

A disease affects 1% of population. Test is 95% accurate for diseased and 90% accurate for healthy. If test is positive, probability of having disease is approximately:

0.95 0.09 0.01 0.50

Q24

A continuous random variable can take:

Only integer values Any value in an interval Only positive values Only finite values

Q25

For constants $a$ and $b$, $E[aX + b] = ?$

$aE[X] + b$ $aE[X]$ $E[X] + b$ $abE[X]$

Q26

$\text{Var}(aX + b) = ?$

$a^2\text{Var}(X)$ $a\text{Var}(X) + b$ $\text{Var}(X) + b$ $a\text{Var}(X)$

Q27

For a binomial distribution $B(n, p)$, the mean is:

$np$ $np(1-p)$ $n(1-p)$ $p(1-p)$

Q28

For a binomial distribution $B(n, p)$, the variance is:

$np$ $np(1-p)$ $n(1-p)$ $p(1-p)$

Q29

A probability distribution must satisfy:

$\sum P(x) = 1$ and $P(x) \geq 0$ $\sum P(x) = 0$ and $P(x) \leq 1$ $\sum P(x) = 1$ and $P(x) \leq 0$ $\sum P(x) = 0$ and $P(x) \geq 1$

Q30

The Bernoulli distribution is a special case of binomial distribution when:

$n = 1$ $p = 0.5$ $n = 0$ $p = 1$

Quiz Results

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