Below are 68 practice questions on Probability (Class 11), sorted Easy → Hard. Tap “Show answer & explanation” under any question to check yourself. Want the full theory first? Read the Probability (Class 11) notes.
📐 Mathematics · Class 11 · JEE
Probability (Class 11) - Practice Questions with Answers
68 free MCQs on Probability (Class 11) with worked answers and explanations. Random experiments, sample space, events, and the axiomatic approach to probability
Take the timed Probability (Class 11) chapterwise test →Easy - 20 questions
Q1.
The set of all possible outcomes of a random experiment is called the:
- A Sample space
- B Outcome pair
- C Trial
- D Event
Show answer & explanation
Answer: A. Sample space
Why: The sample space S is the collection of every possible outcome of the experiment.
Q2.
When a single coin is tossed, the number of elements in the sample space is:
- A 1
- B 2
- C 4
- D 8
Show answer & explanation
Answer: B. 2
Why: S = {H, T}, so n(S) = 2.
Q3.
When two coins are tossed together, the number of possible outcomes is:
- A 2
- B 3
- C 4
- D 8
Show answer & explanation
Answer: C. 4
Why: S = {HH, HT, TH, TT}, giving 4 outcomes.
Q4.
The probability of a sure event is:
- A Undefined
- B 0
- C 1/2
- D 1
Show answer & explanation
Answer: D. 1
Why: By the axioms of probability, P(S) = 1 for the sample space itself.
Q5.
The probability of an impossible event is:
- A 0
- B 1
- C 1/2
- D Negative
Show answer & explanation
Answer: A. 0
Why: The impossible event is the empty set, and P(∅) = 0.
Q6.
The probability of any event A always satisfies:
- A P(A) ≥ 1
- B 0 ≤ P(A) ≤ 1
- C −1 ≤ P(A) ≤ 1
- D P(A) > 1
Show answer & explanation
Answer: B. 0 ≤ P(A) ≤ 1
Why: Probability is a number between 0 and 1 inclusive; values outside this range indicate an error.
Q7.
When a die is rolled once, the probability of getting a 4 is:
- A 1/2
- B 1/3
- C 1/6
- D 4/6
Show answer & explanation
Answer: C. 1/6
Why: There is one favourable outcome out of six equally likely ones, so P = 1/6.
Q8.
When a die is rolled once, the probability of getting an even number is:
- A 2/3
- B 1/6
- C 1/3
- D 1/2
Show answer & explanation
Answer: D. 1/2
Why: The even outcomes 2, 4 and 6 give 3 favourable results out of 6, so P = 3/6 = 1/2.
Q9.
If P(A) = 0.35, then P(A′) equals:
- A 0.65
- B 1.35
- C 0.5
- D 0.35
Show answer & explanation
Answer: A. 0.65
Why: The complement rule gives P(A′) = 1 − P(A) = 1 − 0.35 = 0.65.
Q10.
Two events A and B are mutually exclusive if:
- A P(A) = P(B)
- B A ∩ B = ∅
- C A ∪ B = S
- D A = B
Show answer & explanation
Answer: B. A ∩ B = ∅
Why: Mutually exclusive means the events cannot occur together, so their intersection is empty.
Q11.
An event consisting of exactly one sample point is called a:
- A Sure event
- B Impossible event
- C Simple event
- D Compound event
Show answer & explanation
Answer: C. Simple event
Why: A single-outcome event is called a simple or elementary event.
Q12.
When two dice are thrown, the total number of outcomes is:
- A 72
- B 12
- C 24
- D 36
Show answer & explanation
Answer: D. 36
Why: Each die independently has 6 faces, giving 6 × 6 = 36 ordered outcomes.
Q13.
A card is drawn from a standard pack of 52. The probability that it is a heart is:
- A 1/4
- B 1/2
- C 1/52
- D 1/13
Show answer & explanation
Answer: A. 1/4
Why: There are 13 hearts among 52 cards, so P = 13/52 = 1/4.
Q14.
A card is drawn from a standard pack. The probability that it is a king is:
- A 1/52
- B 1/13
- C 1/4
- D 4/13
Show answer & explanation
Answer: B. 1/13
Why: There are 4 kings in 52 cards, so P = 4/52 = 1/13.
Q15.
When three coins are tossed, the number of elements in the sample space is:
- A 3
- B 6
- C 8
- D 9
Show answer & explanation
Answer: C. 8
Why: Each coin has 2 outcomes, so the total is 2³ = 8.
Q16.
For mutually exclusive events A and B, P(A ∪ B) equals:
- A P(A) × P(B)
- B P(A) − P(B)
- C P(A) + P(B) − 1
- D P(A) + P(B)
Show answer & explanation
Answer: D. P(A) + P(B)
Why: With no overlap to subtract, the addition rule reduces to P(A) + P(B).
Q17.
An event that contains more than one sample point is called a:
- A Compound event
- B Simple event
- C Null event
- D Sure event
Show answer & explanation
Answer: A. Compound event
Why: An event made up of two or more outcomes is a compound event.
Q18.
The complement of the event 'getting a head' when one coin is tossed is:
- A The impossible event
- B Getting a tail
- C Getting a head again
- D The sure event
Show answer & explanation
Answer: B. Getting a tail
Why: With sample space {H, T}, everything that is not a head is a tail.
Q19.
When a die is rolled, the probability of getting a number greater than 6 is:
- A 1
- B 1/2
- C 0
- D 1/6
Show answer & explanation
Answer: C. 0
Why: No face exceeds 6, so this is an impossible event with probability 0.
Q20.
If a bag has 3 red and 2 blue balls, the probability of drawing a red ball is:
- A 1/2
- B 3/2
- C 2/5
- D 3/5
Show answer & explanation
Answer: D. 3/5
Why: There are 3 red balls among 5 total, so P = 3/5.
Medium - 20 questions
Q21.
Two dice are thrown. The probability that the sum is 8 is:
- A 5/36
- B 1/6
- C 4/36
- D 6/36
Show answer & explanation
Answer: A. 5/36
Why: The favourable pairs are (2,6), (3,5), (4,4), (5,3), (6,2) - 5 of the 36 outcomes.
Q22.
Two dice are thrown. The probability of getting a doublet is:
- A 5/36
- B 1/6
- C 1/12
- D 1/36
Show answer & explanation
Answer: B. 1/6
Why: The doublets (1,1) through (6,6) number 6 out of 36, so P = 6/36 = 1/6.
Q23.
Three coins are tossed. The probability of getting at least one head is:
- A 3/8
- B 1/2
- C 7/8
- D 1/8
Show answer & explanation
Answer: C. 7/8
Why: Using the complement, P(no head) = P(TTT) = 1/8, so P(at least one head) = 1 − 1/8 = 7/8.
Q24.
A card is drawn from a pack. The probability that it is a king or a heart is:
- A 17/52
- B 1/13
- C 16/13
- D 4/13
Show answer & explanation
Answer: D. 4/13
Why: P = 4/52 + 13/52 − 1/52 = 16/52 = 4/13; the king of hearts is subtracted once to avoid double counting.
Q25.
If P(A) = 0.4, P(B) = 0.5 and P(A ∩ B) = 0.2, then P(A ∪ B) is:
- A 0.7
- B 0.9
- C 1.1
- D 0.3
Show answer & explanation
Answer: A. 0.7
Why: P(A ∪ B) = 0.4 + 0.5 − 0.2 = 0.7.
Q26.
Two dice are thrown. The probability that the sum is greater than 10 is:
- A 2/9
- B 1/12
- C 1/6
- D 1/9
Show answer & explanation
Answer: B. 1/12
Why: Sums above 10 come from (5,6), (6,5) and (6,6) - 3 outcomes out of 36, so P = 3/36 = 1/12.
Q27.
A number is chosen at random from 1 to 25. The probability that it is a multiple of 5 is:
- A 5/24
- B 4/25
- C 1/5
- D 1/25
Show answer & explanation
Answer: C. 1/5
Why: The multiples 5, 10, 15, 20 and 25 give 5 favourable outcomes from 25, so P = 5/25 = 1/5.
Q28.
A card is drawn from a pack. The probability that it is a face card is:
- A 1/13
- B 4/13
- C 1/4
- D 3/13
Show answer & explanation
Answer: D. 3/13
Why: Jacks, queens and kings across four suits give 12 face cards, so P = 12/52 = 3/13.
Q29.
Two coins are tossed. The probability of getting exactly one head is:
- A 1/2
- B 1/4
- C 3/4
- D 1/3
Show answer & explanation
Answer: A. 1/2
Why: The favourable outcomes HT and TH give 2 of the 4 possibilities, so P = 2/4 = 1/2.
Q30.
If the odds in favour of an event are 3 : 2, the probability of the event is:
- A 2/3
- B 3/5
- C 2/5
- D 3/2
Show answer & explanation
Answer: B. 3/5
Why: Odds of 3 : 2 mean 3 favourable to 2 unfavourable out of 5 total, so P = 3/5.
Q31.
A bag contains 5 red and 7 white balls. One ball is drawn. The probability that it is white is:
- A 7/5
- B 1/2
- C 7/12
- D 5/12
Show answer & explanation
Answer: C. 7/12
Why: There are 7 white among 12 total, so P = 7/12.
Q32.
Two dice are thrown. The probability that the sum is a prime number is:
- A 1/3
- B 7/18
- C 1/2
- D 5/12
Show answer & explanation
Answer: D. 5/12
Why: Prime sums are 2, 3, 5, 7 and 11, occurring 1 + 2 + 4 + 6 + 2 = 15 times out of 36, so P = 15/36 = 5/12.
Q33.
If A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.45, then P(A ∪ B) is:
- A 0.75
- B 0.15
- C 0.135
- D 1
Show answer & explanation
Answer: A. 0.75
Why: For mutually exclusive events the probabilities add directly: 0.3 + 0.45 = 0.75.
Q34.
A letter is chosen at random from the word ASSASSINATION. The probability that it is a vowel is:
- A 1/2
- B 6/13
- C 7/13
- D 5/13
Show answer & explanation
Answer: B. 6/13
Why: The word has 13 letters, of which the vowels A, A, I, A, I, O number 6, so P = 6/13.
Q35.
Two dice are thrown. The probability that both show the same number is:
- A 5/36
- B 1/3
- C 1/6
- D 1/36
Show answer & explanation
Answer: C. 1/6
Why: This is the doublet event, with 6 favourable outcomes out of 36, giving 1/6.
Q36.
If P(A) = 0.6 and P(A ∩ B) = 0.25, the probability that A occurs but B does not is:
- A 0.85
- B 0.25
- C 0.15
- D 0.35
Show answer & explanation
Answer: D. 0.35
Why: P(A ∩ B′) = P(A) − P(A ∩ B) = 0.6 − 0.25 = 0.35.
Q37.
A card is drawn from a pack. The probability that it is neither a heart nor a king is:
- A 9/13
- B 4/13
- C 3/13
- D 1/2
Show answer & explanation
Answer: A. 9/13
Why: P(heart or king) = 16/52, so the complement is 36/52 = 9/13.
Q38.
Two dice are thrown. The probability that the sum is 7 is:
- A 1/12
- B 1/6
- C 5/36
- D 1/9
Show answer & explanation
Answer: B. 1/6
Why: The six pairs (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) give P = 6/36 = 1/6 - the most likely sum.
Q39.
One number is chosen from 1 to 100. The probability that it is a perfect square is:
- A 1/50
- B 9/100
- C 1/10
- D 1/100
Show answer & explanation
Answer: C. 1/10
Why: The perfect squares 1, 4, 9, …, 100 number 10, so P = 10/100 = 1/10.
Q40.
If P(A) = 0.5, P(B) = 0.4 and P(A ∪ B) = 0.7, then P(A ∩ B) is:
- A 0.1
- B 0.3
- C 0.9
- D 0.2
Show answer & explanation
Answer: D. 0.2
Why: Rearranging the addition rule: P(A ∩ B) = 0.5 + 0.4 − 0.7 = 0.2.
Hard - 28 questions
Q41.
Why is 'mutually exclusive' not the same as 'exhaustive'?
- A Mutually exclusive means the events cannot overlap; exhaustive means together they cover the whole sample space
- B The two terms describe identical conditions expressed in different words
- C Mutually exclusive events must always be equally likely as well
- D Exhaustive events must always have equal probabilities in every case
Show answer & explanation
Answer: A. Mutually exclusive means the events cannot overlap; exhaustive means together they cover the whole sample space
Why: A ∩ B = ∅ concerns overlap, while A ∪ B = S concerns coverage; events can satisfy either condition without the other.
Q42.
When two dice are rolled, why is treating (2, 3) and (3, 2) as a single outcome an error?
- A Unordered pairs are correct, but the arithmetic becomes harder to manage
- B The two dice are distinguishable, so the resulting 21 unordered pairs are not equally likely
- C Ordered pairs are only used when the dice have different numbers of faces
- D The sample space must always contain exactly 36 elements by definition
Show answer & explanation
Answer: B. The two dice are distinguishable, so the resulting 21 unordered pairs are not equally likely
Why: Each ordered pair has probability 1/36, so an unordered pair like {2,3} has probability 2/36 while {4,4} has only 1/36 - collapsing them breaks the equally-likely assumption.
Q43.
The formula P(A) = n(A)/n(S) is valid only when:
- A The event A is a simple event containing one outcome
- B The events under consideration are mutually exclusive
- C All outcomes in the sample space are equally likely
- D The sample space contains a finite number of outcomes only
Show answer & explanation
Answer: C. All outcomes in the sample space are equally likely
Why: Counting favourable outcomes assumes each carries the same weight; for a biased die or unequal outcomes the axiomatic definition must be used instead.
Q44.
Four cards are drawn from a pack. The probability that all four are kings is:
- A 1/13²
- B 4/52
- C 1/2704
- D 1/270725
Show answer & explanation
Answer: D. 1/270725
Why: The count is C(4,4)/C(52,4) = 1/270725, since there is exactly one way to choose all four kings.
Q45.
Three cards are drawn from a pack. The probability that all three are hearts is:
- A 11/850
- B 1/64
- C 13/52
- D 3/13
Show answer & explanation
Answer: A. 11/850
Why: C(13,3)/C(52,3) = 286/22100 = 11/850.
Q46.
If A and B are events with P(A) = 0.5, P(B) = 0.3, what is the largest possible value of P(A ∩ B)?
- A 0.15
- B 0.3
- C 0.5
- D 0.8
Show answer & explanation
Answer: B. 0.3
Why: The intersection cannot exceed either event, so P(A ∩ B) ≤ min(0.5, 0.3) = 0.3, attained when B ⊆ A.
Q47.
Two dice are thrown. The probability that the sum is neither 7 nor 11 is:
- A 1/6
- B 5/6
- C 7/9
- D 2/9
Show answer & explanation
Answer: C. 7/9
Why: Sum 7 occurs 6 ways and sum 11 occurs 2 ways, so 8/36 = 2/9 is excluded, leaving 1 − 2/9 = 7/9.
Q48.
A committee of 3 is chosen from 5 men and 4 women. The probability that it has exactly 2 women is:
- A 3/14
- B 1/2
- C 10/21
- D 5/14
Show answer & explanation
Answer: D. 5/14
Why: Favourable = C(4,2)·C(5,1) = 6 × 5 = 30, total = C(9,3) = 84, so P = 30/84 = 5/14.
Q49.
Why does the addition rule subtract P(A ∩ B)?
- A Outcomes lying in both events would otherwise be counted twice in P(A) + P(B)
- B The intersection always has probability zero and must be removed
- C Subtraction converts the union into a mutually exclusive pair of events
- D Probabilities must sum to exactly one across all events considered
Show answer & explanation
Answer: A. Outcomes lying in both events would otherwise be counted twice in P(A) + P(B)
Why: P(A) and P(B) each include the shared outcomes, so the overlap is counted twice and must be removed once.
Q50.
Two cards are drawn without replacement. The probability that both are aces is:
- A 4/52
- B 1/221
- C 1/169
- D 1/13
Show answer & explanation
Answer: B. 1/221
Why: C(4,2)/C(52,2) = 6/1326 = 1/221; equivalently (4/52)(3/51).
Q51.
Six people sit in a row at random. The probability that two particular people sit together is:
- A 2/5
- B 1/2
- C 1/3
- D 1/6
Show answer & explanation
Answer: C. 1/3
Why: Treating the pair as one block gives 5!·2! = 240 arrangements out of 6! = 720, so P = 240/720 = 1/3.
Q52.
If the odds against an event are 5 : 3, the probability of the event is:
- A 5/8
- B 3/5
- C 5/3
- D 3/8
Show answer & explanation
Answer: D. 3/8
Why: Odds against of 5 : 3 mean 3 favourable to 5 unfavourable, so P = 3/(5 + 3) = 3/8.
Q53.
A number is chosen from 1 to 20. The probability that it is divisible by 3 or 5 is:
- A 9/20
- B 1/2
- C 2/5
- D 11/20
Show answer & explanation
Answer: A. 9/20
Why: Multiples of 3 number 6, of 5 number 4, and of 15 number 1, so by inclusion–exclusion 6 + 4 − 1 = 9, giving 9/20.
Q54.
Why is P(at least one) usually computed via the complement?
- A The complement always produces a larger and therefore safer probability value
- B The complement 'none' is a single scenario, while 'at least one' splits into many separate cases
- C The complement rule is the only exact method available for such events
- D 'At least one' events are always impossible to count directly in principle
Show answer & explanation
Answer: B. The complement 'none' is a single scenario, while 'at least one' splits into many separate cases
Why: Counting 'exactly one', 'exactly two' and so on is laborious, whereas 'none' is a single case, so 1 − P(none) is far quicker and less error-prone.
Q55.
Three dice are thrown. The probability that all three show the same number is:
- A 1/6
- B 6/216
- C 1/36
- D 1/216
Show answer & explanation
Answer: C. 1/36
Why: There are 6 favourable triples out of 6³ = 216, giving 6/216 = 1/36.
Q56.
Two events A and B satisfy P(A) = 0.4, P(B) = 0.7 and P(A ∪ B) = 1. What is P(A ∩ B)?
- A 0.3
- B 0.28
- C 0
- D 0.1
Show answer & explanation
Answer: D. 0.1
Why: P(A ∩ B) = P(A) + P(B) − P(A ∪ B) = 0.4 + 0.7 − 1 = 0.1.
Q57.
From a pack of 52, one card is drawn. The probability that it is a red face card is:
- A 3/26
- B 3/13
- C 1/26
- D 6/13
Show answer & explanation
Answer: A. 3/26
Why: Hearts and diamonds contribute 3 face cards each, so 6 out of 52, giving 6/52 = 3/26.
Q58.
Four coins are tossed. The probability of getting exactly two heads is:
- A 5/16
- B 3/8
- C 1/4
- D 1/2
Show answer & explanation
Answer: B. 3/8
Why: C(4,2) = 6 favourable outcomes out of 2⁴ = 16, so P = 6/16 = 3/8.
Q59.
If A ⊆ B, then which relation must hold?
- A P(A) = P(B) always
- B P(A) + P(B) = 1
- C P(A) ≤ P(B)
- D P(A) ≥ P(B)
Show answer & explanation
Answer: C. P(A) ≤ P(B)
Why: Every outcome in A also lies in B, so B carries at least as much probability - monotonicity of probability.
Q60.
Two dice are thrown. Given the sample space of 36 equally likely outcomes, the sum with the highest probability is:
- A 6
- B 8
- C 12
- D 7
Show answer & explanation
Answer: D. 7
Why: Sum 7 arises in 6 ways, more than any other total, because it is the centre of the symmetric distribution of sums.
Q61.
A card is drawn from a standard deck. The probability it is a face card is:
- A 3/13
- B 1/13
- C 4/13
- D 12/13
Show answer & explanation
Answer: A. 3/13
Why: There are 12 face cards, so the probability is 12/52 = 3/13.
Q62.
Two fair dice are thrown. The probability that the sum is a prime number is:
- A 5/12
- B 1/3
- C 1/2
- D 7/18
Show answer & explanation
Answer: A. 5/12
Why: Prime sums 2, 3, 5, 7, 11 occur in 1 + 2 + 4 + 6 + 2 = 15 ways, giving 15/36 = 5/12.
Q63.
The letters A, B, C, D are arranged at random in a row. The probability that A comes before B is:
- A 1/4
- B 1/2
- C 1/3
- D 3/4
Show answer & explanation
Answer: B. 1/2
Why: By symmetry A is before B in exactly half the arrangements, so the probability is 1/2.
Q64.
Two cards are drawn from a standard deck without replacement. The probability that both are aces is:
- A 1/221
- B 1/169
- C 2/221
- D 1/13
Show answer & explanation
Answer: A. 1/221
Why: C(4,2)/C(52,2) = 6/1326 = 1/221.
Q65.
If A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.4, then P(A ∪ B) equals:
- A 0.12
- B 0.7
- C 0.1
- D 1
Show answer & explanation
Answer: B. 0.7
Why: For mutually exclusive events P(A ∪ B) = 0.3 + 0.4 = 0.7.
Q66.
If the odds in favour of an event are 3 : 2, the probability of the event is:
- A 2/5
- B 3/5
- C 3/2
- D 2/3
Show answer & explanation
Answer: B. 3/5
Why: Probability = 3/(3 + 2) = 3/5.
Q67.
A number is chosen at random from 1 to 20. The probability it is divisible by 3 or 5 is:
- A 9/20
- B 1/2
- C 7/20
- D 11/20
Show answer & explanation
Answer: A. 9/20
Why: Divisible by 3: 6 numbers, by 5: 4, by 15: 1, so 6 + 4 − 1 = 9 out of 20.
Q68.
Three fair coins are tossed. The probability of getting at least two tails is:
- A 3/8
- B 1/2
- C 1/4
- D 5/8
Show answer & explanation
Answer: B. 1/2
Why: P(2 tails) + P(3 tails) = 3/8 + 1/8 = 1/2.