68 free MCQs on Probability, each with its own worked answer and explanation. Chance, events, conditional probability, and Bayes theorem
Take the timed Probability chapterwise test →68 practice questions on Probability, sorted Easy → Hard. Try each one first, then open its answer page for the worked explanation. Want the full theory first? Read the Probability notes.
Venn diagram of the universal set U with events A and B, showing the intersection (A and B), the parts unique to each event, and the complement region outside both.
Q1.
Probability of an event ranges from:
Q8.
P(drawing a king from a deck of 52 cards) =
Q12.
Two events that cannot occur simultaneously are:
Q15.
If events A and B are mutually exclusive, P(A or B) =
Q17.
P(getting a tail when a coin is tossed 3 times, all tails) =
Q21.
P(A ∪ B) = P(A) + P(B) - P(A ∩ B). If P(A) = 0.3, P(B) = 0.4, P(A ∩ B) = 0.1, find P(A ∪ B).
Q22.
Conditional probability P(A|B) =
Q23.
A and B are independent events with P(A)=0.4, P(B)=0.5. P(A ∩ B) =
Q24.
A bag has 4 red and 6 blue balls. Two balls drawn without replacement. P(both red) =
Q25.
Bayes' theorem is used to:
Q26.
Binomial distribution B(n,p) gives probability for:
Q29.
From a pack of 52 cards, P(king or queen) =
Q31.
If P(A) = 1/3 and A,B are mutually exclusive with P(A ∪ B) = 1/2, then P(B) =
Q32.
A fair coin is tossed 4 times. P(exactly 2 heads) =
Q33.
The total probability theorem states:
Q34.
Poisson distribution is used for:
Q35.
Three cards are drawn from 52 without replacement. P(all aces) =
Q36.
If events A and B are exhaustive, then:
Q37.
The geometric distribution models:
Q38.
If P(A) = 0.6, P(A|B) = 0.5, P(B) = 0.4, then P(A ∩ B) =
Q39.
For independent events A and B, P(A|B) =
Q40.
A box has 5 good and 3 defective bulbs. P(2nd is defective given 1st was good) =
Q41.
Using Bayes' theorem: prior P(H) = 0.4, P(E|H) = 0.9, P(E|not H) = 0.3. Find P(H|E).
Q42.
In a Poisson distribution with lambda = 3, P(X = 2) =
Q43.
Random variable X has E[X] = 3 and E[X²] = 13. Variance of X =
Q44.
For jointly distributed random variables X,Y: Cov(X,Y) = 0 implies:
Q45.
Negative Binomial distribution gives probability of:
Q46.
The characteristic function of a distribution is related to the MGF by substituting:
Q47.
For n large, Poisson(lambda) can approximate Binomial B(n,p) when:
Q48.
E[aX + bY] =
Q49.
The strong law of large numbers states:
Q50.
If X ~ N(μ,σ²), the standard normal variable Z =
Q51.
Var(X + Y) when X and Y are not independent:
Q52.
The probability that at least one event occurs: P(A ∪ B) = 1 - P(A' ∩ B') by:
Q53.
In a Markov chain, the transition probability P(i,j) represents:
Q54.
The expectation E[X²] is also called:
Q55.
If X₁,...,Xₙ are iid with mean μ and variance σ², the central limit theorem says X̄ has approx distribution:
Q56.
In hypothesis testing, type I error is:
Q57.
The convolution of two independent distributions gives:
Q58.
For a discrete random variable X: sum of all p(x) over all x must equal:
Q59.
Exponential distribution is the only continuous distribution with the memoryless property. P(X > s + t | X > s) =
Q60.
A fair die is rolled once. The probability of getting an even number is:
Q61.
When two fair dice are thrown, the probability that the sum is 8 is:
Q62.
The probability of getting at least one head in three tosses of a fair coin is:
Q63.
A bag has 4 red and 6 black balls. Two are drawn without replacement. The probability both are red is:
Q64.
Box 1 has 2 white and 3 black balls; Box 2 has 4 white and 1 black. A box is chosen at random and a white ball is drawn. The probability it came from Box 1 is:
Q65.
A card is drawn from a standard 52-card deck. The probability it is a king or a heart is:
Q66.
If P(A ∩ B) = 0.2 and P(B) = 0.5, then P(A | B) equals:
Q67.
Three fair coins are tossed. The probability of exactly two heads is:
Q68.
In 5 independent trials with success probability 1/2, the probability of exactly 3 successes is:
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