Probability & Permutations MCQs With Answers
Probability questions reward a clear method more than heavy calculation. These probability and permutation MCQs cover coins, dice and playing cards, drawing coloured balls with and without replacement, at least one head, complementary events, arrangements of letters in words such as APPLE and LEADER, circular seating, selecting committees with combinations and counting subsets. They are useful for NTS, GAT, PPSC and FPSC tests and for students revising the probability chapter.
Take a scored quiz instead →What is the probability of picking an even number from 1 to 10?
2, 4, 6, 8, 10 (5 numbers). 5/10 = 0.5.
A bag has 3 white, 2 black balls. Probability of white?
3 white / 5 total.
The probability of a leap year having 53 Sundays is:
366 days = 52 weeks + 2 days. 2/7 chance.
Two fair coins are tossed together. What is the probability of getting exactly one head?
The four equally likely outcomes are HH, HT, TH and TT.
Exactly one head occurs in HT and TH, so the probability = 2/4 = 1/2.
A bag contains 5 red and 3 blue balls. Probability of picking a red ball?
5 red / 8 total.
Three fair coins are tossed. What is the probability of getting at least one tail?
There are 2³ = 8 equally likely outcomes, and only one of them (HHH) has no tail.
So P(at least one tail) = 1 − 1/8 = 7/8.
What is the probability of picking a King from a standard deck of 52 cards?
4/52 = 1/13.
Probability of drawing an Ace from a deck of 52 cards?
4/52 = 1/13.
A bag contains 5 red, 3 blue, and 2 green balls. Probability of drawing a blue ball?
3/(5+3+2) = 3/10 = 0.3.
In how many ways can the word ‘APPLE’ be arranged?
5! / 2! (P repeats) = 120 / 2 = 60.
What is the probability of drawing a red card from a well-shuffled standard deck of 52 cards?
- A standard deck holds 52 cards, and 26 of them are red: 13 hearts and 13 diamonds.
- Probability = favourable outcomes ÷ total outcomes = 26/52 = 1/2.
- The remaining 26 cards are black spades and clubs, so red and black are equally likely in a single draw.
- Option A 1/4 is the probability of drawing one particular suit, such as hearts on their own.
- Option C 1/13 is the probability of drawing one particular rank, such as a king, not a colour.
Box has 3 black, 7 white. Prob white?
7/10=0.7.
In how many ways can 4 different books be arranged on a shelf?
4! = 4*3*2*1 = 24.
Probability of drawing a black card from a pack?
26 black cards / 52 total = 1/2.
Box has 4 red, 6 white. Prob of red?
4/10 = 0.4.
A dice is rolled. Probability of getting a number greater than 4?
Numbers > 4 are 5, 6. Prob = 2/6 = 1/3.
How many ways can 3 people be seated in 5 chairs?
P(5,3) = 5 * 4 * 3 = 60.
Probability of getting sum 7 with 2 dice?
Pairs: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1). 6/36 = 1/6.
Box has 5 blue, 5 red. Prob of blue?
5/10=0.5.
A bag contains 3 green and 5 yellow balls. What is the probability of picking a green ball?
Total balls = 3 + 5 = 8. Favorable = 3. Probability = 3/8.
How many arrangements are possible for the letters of ‘BOOK’?
Total letters = 4. 'O' repeats twice. Arrangements = 4! / 2! = 24 / 2 = 12.
Probability of drawing a diamond from a deck?
13 / 52 = 1/4.
Box has 2 red, 8 blue. Prob of red?
2 / 10 = 0.2.
A bag contains 6 red and 4 blue balls. If two balls are picked, what is the probability that both are red?
P = (6/10) * (5/9) = 30 / 90 = 1/3.
Probability of drawing a spade from a deck?
13 spades out of 52 cards = 13/52 = 1/4.
Box has 3 black, 7 white. Prob of black?
3 / 10 = 0.3.
A bag contains 5 red and 3 blue balls. If two balls are drawn, what is the probability that one is red and one is blue?
Total = C(8,2) = 28. Favorable = C(5,1)*C(3,1) = 5*3 = 15. Prob = 15/28.
How many ways can 5 books be arranged on a shelf?
- Arranging 5 different books in a row is 5! = 5 × 4 × 3 × 2 × 1 = 120.
- Think of the slots: the first has 5 choices, the next 4, then 3, then 2, then 1, and multiplying gives 120.
- Order matters on a shelf, so this is a permutation rather than a combination.
- Option A 24 is 4!, which would be the answer for only 4 books.
- If two books had to stay together, treat them as one unit and the count drops to 4! × 2 = 48.
A bag has 3 black, 4 white balls. Probability of picking 2 white balls?
- Choosing 2 white balls from the 4 available gives C(4,2) = 6 ways.
- Choosing any 2 balls from all 7 gives C(7,2) = 21 ways.
- So the probability is 6/21, which simplifies to 2/7.
- Order does not matter when two balls are drawn together, so combinations are used instead of permutations.
- The step-by-step method agrees: (4/7) × (3/6) = 12/42 = 2/7, because one white ball is gone before the second draw.
Probability of drawing an Ace from deck?
4/52 = 1/13.
Ball selection: 5 red, 4 blue. Probability not red?
- Total balls = 5 + 4 = 9, and "not red" refers to the 4 blue balls.
- So the probability is 4 out of 9, written as 4/9.
- A second route is the complement rule: P(not red) = 1 − P(red) = 1 − 5/9 = 4/9.
- Option B 5/9 is the probability of drawing a red ball, so the word "not" must be read carefully.
- The two probabilities add up to 1, which gives you a quick way to check the answer.
Two cards are drawn at random from a standard deck of 52 cards without replacement. What is the probability that both are aces?
Probability = C(4,2) / C(52,2) = 6 / 1326 = 1 / 221.
In how many different ways can the letters of the word ‘MATHS’ be arranged?
Total letters = 5 (all distinct). Total arrangements = 5! = 120.
A bag contains 4 red, 5 green, and 6 yellow balls. A ball is drawn at random. What is the probability that it is green or yellow?
Total balls = 4 + 5 + 6 = 15. Favorable balls (green or yellow) = 5 + 6 = 11. Probability = 11 / 15.
Two dice are thrown together. What is the probability that the sum of the numbers on the top faces is divisible by 5?
Favorable sums: 5 and 10. Pairs for sum 5: (1,4), (2,3), (3,2), (4,1) -> 4. Pairs for sum 10: (4,6), (5,5), (6,4) -> 3. Total favorable = 7. Probability = 7 / 36.
A box contains 6 red, 4 white, and 8 blue balls. If one ball is drawn at random, what is the probability that it is NOT white?
Total balls = 6 + 4 + 8 = 18. Non-white balls = 6 + 8 = 14. Probability = 14 / 18 = 7/9.
In how many ways can 6 people be seated around a circular table?
Number of circular permutations of n distinct items = (n - 1)!. For n = 6: (6 - 1)! = 5! = 120.
From a deck of 52 cards, two cards are drawn at random. What is the probability that both are Kings?
Probability = C(4, 2) / C(52, 2) = 6 / 1326 = 1 / 221.
A fair die is thrown once. What is the probability of getting a prime number?
Prime numbers on a die = {2, 3, 5} (3 outcomes). Total outcomes = 6. Probability = 3 / 6 = 1/2.
A box contains 3 red, 4 white, and 5 blue balls. If one ball is drawn at random, what is the probability that it is NOT red?
Total balls = 3 + 4 + 5 = 12. Non-red balls = 4 + 5 = 9. Probability = 9 / 12 = 3/4.
A bag contains 3 red, 5 green, and 4 blue balls. If two balls are drawn at random together, what is the probability that both are green?
Favorable ways = C(5, 2) = 10. Total ways = C(12, 2) = 66. Probability = 10 / 66 = 5/33.
In how many ways can 5 boys and 3 girls be seated in a row such that all 3 girls are always together?
Treat 3 girls as 1 unit. Total units = 5 boys + 1 girl-unit = 6 units. Arranged in 6! ways. 3 girls can be arranged among themselves in 3! ways. Total = 6! * 3! = 720 * 6 = 4320.
A card is drawn from a well-shuffled deck of 52 cards. What is the probability that it is a Face Card (Jack, Queen, King)?
Total face cards = 4 Jacks + 4 Queens + 4 Kings = 12. Probability = 12 / 52 = 3/13.
In a simultaneous throw of two dice, what is the probability of getting a total of at least 10?
Favorable outcomes for sum >= 10: Sum 10 (4,6; 5,5; 6,4), Sum 11 (5,6; 6,5), Sum 12 (6,6). Total = 3 + 2 + 1 = 6. Probability = 6 / 36 = 1/6.
A box contains 5 red and 5 blue balls. Two balls are drawn one by one WITH replacement. What is the probability that both are red?
P(1st Red) = 5/10 = 1/2. With replacement, P(2nd Red) = 5/10 = 1/2. Combined P = 1/2 * 1/2 = 1/4.
Two dice are thrown simultaneously. What is the probability of getting a sum of 8?
Favorable outcomes for sum 8: (2,6), (3,5), (4,4), (5,3), (6,2) -> 5 outcomes. Total possible outcomes = 36. Probability = 5/36.
In how many different ways can the letters of the word ‘LEADER’ be arranged?
Total letters = 6, with 'E' repeating 2 times. Total arrangements = 6! / 2! = 720 / 2 = 360.
What is the probability of getting at least one head when three unbiased coins are tossed together?
Total outcomes = 2^3 = 8. Outcome with NO head = (T, T, T) [1 outcome]. Probability of at least one head = 1 - 1/8 = 7/8.
A bag contains 5 red, 3 blue, and 2 green balls. If one ball is drawn at random, what is the probability that it is NOT blue?
Total balls = 5 + 3 + 2 = 10. Non-blue balls = 5 + 2 = 7. Probability = 7/10.
If C(n, 2) = 45, find the value of n.
C(n, 2) = n(n-1) / 2 = 45 => n(n-1) = 90 => n^2 - n - 90 = 0 => (n - 10)(n + 9) = 0 => n = 10.
A box contains 4 red cards and 6 black cards. Two cards are drawn one after another without replacement. What is the probability that both are red?
Probability of 1st red = 4/10 = 2/5. Probability of 2nd red = 3/9 = 1/3. Combined probability = (2/5) * (1/3) = 2/15.
A die is rolled once. What is the probability of getting an even prime number?
The only even prime number on a standard die is 2 (1 favorable outcome out of 6). Probability = 1/6.
Two dice are thrown simultaneously. What is the probability of getting a double (same number on both)?
Favorable outcomes: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) -> 6 outcomes. Total outcomes = 36. Probability = 6 / 36 = 1/6.
What is the probability of picking a prime number from numbers 1 to 10 inclusive at random?
Prime numbers between 1 and 10 are 2, 3, 5, 7 (total 4). Total outcomes = 10. Probability = 4 / 10 = 2/5.
A bag contains 5 red, 3 blue, and 2 yellow balls. Two balls are drawn one after another without replacement. What is the probability that both are red?
Total balls = 10. P(first red) = 5/10 = 1/2. P(second red) = 4/9. Combined probability = (1/2) * (4/9) = 2/9.
In how many ways can 4 letters be selected from 6 different letters?
Combination C(6, 4) = 6! / (4! * 2!) = (6 * 5) / 2 = 15.
What is the probability of getting a sum of 10 when rolling two fair dice?
Favorable outcomes: (4,6), (5,5), (6,4) -> 3 outcomes. Total outcomes = 36. Probability = 3/36 = 1/12.
In how many different ways can 5 people be seated in a row?
Number of arrangements = 5! = 5 * 4 * 3 * 2 * 1 = 120.
A card is drawn from a pack of 52 cards. What is the probability of getting an Ace or a King?
Number of Aces = 4, Kings = 4. Total favorable = 8. Probability = 8/52 = 2/13.
A bag contains 4 red, 5 blue, and 6 green balls. A ball is drawn at random. What is the probability it is blue?
Total balls = 4 + 5 + 6 = 15. Blue balls = 5. Probability = 5/15 = 1/3.
How many subsets does a set containing 4 elements have?
Number of subsets of a set with n elements is 2^n. Here 2^4 = 16.
In how many different ways can the letters of the word ‘MATH’ be arranged?
Total distinct letters = 4. Number of permutations = 4! = 4 * 3 * 2 * 1 = 24 ways.
A single die is rolled. What is the probability of getting an even number?
Favorable outcomes (2, 4, 6) = 3. Total possible outcomes = 6. Probability = 3 / 6 = 1/2.
A bag contains 3 red, 4 white, and 5 blue balls. Two balls are drawn at random. What is the probability that both are red?
Total balls = 12. Total ways to pick 2 = 12C2 = 66. Ways to pick 2 red = 3C2 = 3. Probability = 3 / 66 = 1/22.
A bag contains 4 red balls, 5 blue balls, and 3 green balls. If one ball is drawn at random, what is the probability that it is NOT blue?
Total balls = 4 + 5 + 3 = 12. Non-blue balls = 4 red + 3 green = 7. Probability of drawing a non-blue ball = Favorable outcomes / Total outcomes = 7/12.
What is the probability of getting at least one head when two fair coins are tossed?
Outcomes = {HH, HT, TH, TT}. Favorable outcomes for at least 1 head = {HH, HT, TH} (3 outcomes). Probability = 3/4.
How many different 3-letter words (with or without meaning) can be formed using letters of ‘MATH’ without repetition?
Permutation 4P3 = 4! / (4 - 3)! = 24 / 1 = 24.
In how many ways can a committee of 3 people be selected from 6 people?
Combination 6C3 = 6! / (3! * 3!) = (6 * 5 * 4) / (3 * 2 * 1) = 20.
A bag contains 4 red balls and 6 blue balls. If one ball is drawn at random, what is the probability that it is red?
Total balls = 4 + 6 = 10. Probability of drawing a red ball = 4 / 10 = 2/5.