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Probability

Coins, dice, cards, balls and basic probability rules.

45 solved questions · 15 easy · 20 medium · 10 hard

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Shortcuts & formulas

  • P(E) = favourable outcomes / total outcomes; 0 <= P(E) <= 1.
  • P(not E) = 1 - P(E). 'At least one' = 1 - P(none).
  • P(A or B) = P(A) + P(B) - P(A and B); for mutually exclusive events P(A and B) = 0.
  • Independent events: P(A and B) = P(A) x P(B).
  • Conditional probability: P(A|B) = P(A and B)/P(B).
  • Two dice: 36 outcomes; sum 7 has 6 ways (most likely). Three coins: 8 outcomes. Pack: 52 cards = 26 red + 26 black, 13 of each suit, 12 face cards.
  • Without replacement: multiply the changing fractions or use combinations (nCr/nCr).
  • Odds in favour = P : (1 - P).

Solved examples

Try each one first, then open the answer and the step-by-step solution.

Easy questions

Q1.

A fair die is thrown once. What is the probability of getting a number greater than 4?

  1. A1/3
  2. B1/2
  3. C1/6
  4. D2/3
Show answer & solution

Answer: A. 1/3

Solution

Favourable: 5, 6 -> 2 outcomes Total outcomes = 6 P = 2/6 = 1/3 Answer: 1/3

Q2.

A fair die is rolled. What is the probability that the number is a prime number?

  1. A1/3
  2. B1/2
  3. C2/3
  4. D1/6
Show answer & solution

Answer: B. 1/2

Solution

Primes on a die: 2, 3, 5 -> 3 outcomes P = 3/6 = 1/2 Answer: 1/2

Q3.

One card is drawn at random from a well-shuffled pack of 52 cards. What is the probability that it is a king?

  1. A1/4
  2. B1/52
  3. C1/13
  4. D4/13
Show answer & solution

Answer: C. 1/13

Solution

There are 4 kings in 52 cards. P = 4/52 = 1/13 Answer: 1/13

Q4.

One card is drawn from a pack of 52 cards. What is the probability that it is a red face card (jack, queen or king of hearts or diamonds)?

  1. A3/13
  2. B3/52
  3. C1/13
  4. D3/26
Show answer & solution

Answer: D. 3/26

Solution

Face cards per suit = 3; red suits = 2 -> 6 red face cards. P = 6/52 = 3/26 Answer: 3/26

Q5.

A card is drawn at random from a pack of 52 cards. What is the probability that it is a heart?

  1. A1/4
  2. B1/13
  3. C1/2
  4. D1/52
Show answer & solution

Answer: A. 1/4

Solution

There are 13 hearts in 52 cards. P = 13/52 = 1/4 Answer: 1/4

Q6.

A bag contains 5 red, 3 blue and 2 green balls. One ball is drawn at random. What is the probability that it is red?

  1. A5/8
  2. B1/2
  3. C3/10
  4. D1/5
Show answer & solution

Answer: B. 1/2

Solution

Total balls = 5 + 3 + 2 = 10 P(red) = 5/10 = 1/2 Answer: 1/2

Q7.

Two fair coins are tossed together. What is the probability of getting two heads?

  1. A1/2
  2. B1/3
  3. C1/4
  4. D3/4
Show answer & solution

Answer: C. 1/4

Solution

Outcomes: HH, HT, TH, TT -> 4 equally likely. P(HH) = 1/4 Answer: 1/4

Q8.

Two fair dice are thrown together. What is the probability that the sum of the numbers is 7?

  1. A1/12
  2. B7/36
  3. C1/36
  4. D1/6
Show answer & solution

Answer: D. 1/6

Solution

Pairs with sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 Total = 36 P = 6/36 = 1/6 Answer: 1/6

Q9.

A number is chosen at random from 1 to 20. What is the probability that it is a multiple of 3?

  1. A3/10
  2. B7/20
  3. C1/3
  4. D1/4
Show answer & solution

Answer: A. 3/10

Solution

Multiples of 3 up to 20: 3, 6, 9, 12, 15, 18 = 6 P = 6/20 = 3/10 Answer: 3/10

Q10.

The probability that it rains tomorrow is 7/20. What is the probability that it does not rain?

  1. A7/13
  2. B13/20
  3. C7/20
  4. D3/20
Show answer & solution

Answer: B. 13/20

Solution

P(not rain) = 1 - 7/20 = 13/20 Answer: 13/20

Q11.

A letter is chosen at random from the word CHENNAI. What is the probability that it is a vowel?

  1. A4/7
  2. B2/7
  3. C3/7
  4. D1/2
Show answer & solution

Answer: C. 3/7

Solution

Letters: C, H, E, N, N, A, I = 7; vowels E, A, I = 3 P = 3/7 Answer: 3/7

Q12.

Three fair coins are tossed. What is the probability of getting all tails?

  1. A1/3
  2. B1/6
  3. C3/8
  4. D1/8
Show answer & solution

Answer: D. 1/8

Solution

Total outcomes = 2^3 = 8; all tails = TTT only. P = 1/8 Answer: 1/8

Q13.

A number is selected at random from 1 to 25. What is the probability that it is a prime number?

  1. A9/25
  2. B8/25
  3. C2/5
  4. D11/25
Show answer & solution

Answer: A. 9/25

Solution

Primes up to 25: 2, 3, 5, 7, 11, 13, 17, 19, 23 = 9 P = 9/25 Answer: 9/25

Q14.

Two fair dice are thrown. What is the probability that the sum is 12?

  1. A1/12
  2. B1/36
  3. C1/18
  4. D1/6
Show answer & solution

Answer: B. 1/36

Solution

Only (6,6) gives 12. P = 1/36 Answer: 1/36

Q15.

The probability of winning a match is 3/8. What are the odds in favour of winning?

  1. A3 : 8
  2. B5 : 3
  3. C3 : 5
  4. D8 : 3
Show answer & solution

Answer: C. 3 : 5

Solution

P(win) = 3/8, so P(not win) = 5/8. Odds in favour = 3/8 : 5/8 = 3 : 5 Answer: 3 : 5

Medium questions

Q16.

Two cards are drawn one after another, without replacement, from a pack of 52 cards. What is the probability that both are kings?

  1. A1/169
  2. B1/26
  3. C4/663
  4. D1/221
Show answer & solution

Answer: D. 1/221

Solution

P(first king) = 4/52 P(second king | first king) = 3/51 P = (4/52) x (3/51) = 12/2652 = 1/221 Answer: 1/221

Shortcut: 4C2/52C2 = 6/1326 = 1/221

Q17.

A bag has 6 blue and 4 white balls. Two balls are drawn at random together. What is the probability that both are of the same colour?

  1. A7/15
  2. B8/15
  3. C13/25
  4. D2/15
Show answer & solution

Answer: A. 7/15

Solution

Total ways = 10C2 = 45 Both blue = 6C2 = 15; both white = 4C2 = 6 P = (15 + 6)/45 = 21/45 = 7/15 Answer: 7/15

Q18.

Three fair coins are tossed. What is the probability of getting exactly two heads?

  1. A1/8
  2. B3/8
  3. C1/2
  4. D1/4
Show answer & solution

Answer: B. 3/8

Solution

Total outcomes = 8 Exactly 2 heads: HHT, HTH, THH = 3 P = 3/8 Answer: 3/8

Q19.

A fair coin is tossed 4 times. What is the probability of getting at least one head?

  1. A1/16
  2. B1/2
  3. C15/16
  4. D7/8
Show answer & solution

Answer: C. 15/16

Solution

P(no head) = (1/2)^4 = 1/16 P(at least one head) = 1 - 1/16 = 15/16 Answer: 15/16

Q20.

Asha and Ravi attempt a problem independently. Their chances of solving it are 1/3 and 1/4 respectively. What is the probability that the problem is solved?

  1. A1/12
  2. B7/12
  3. C2/3
  4. D1/2
Show answer & solution

Answer: D. 1/2

Solution

P(neither solves) = (1 - 1/3)(1 - 1/4) = (2/3)(3/4) = 1/2 P(solved) = 1 - 1/2 = 1/2 Answer: 1/2

Q21.

A die is thrown. Given that the number is greater than 3, what is the probability that it is even?

  1. A2/3
  2. B1/2
  3. C1/3
  4. D3/4
Show answer & solution

Answer: A. 2/3

Solution

Given > 3: outcomes 4, 5, 6 (3 outcomes). Even among them: 4, 6 (2 outcomes). P = 2/3 Answer: 2/3

Q22.

A number is chosen at random from 1 to 100. What is the probability that it is divisible by 4 or 6?

  1. A41/100
  2. B33/100
  3. C2/5
  4. D1/4
Show answer & solution

Answer: B. 33/100

Solution

Divisible by 4: 25; by 6: 16; by both (multiples of 12): 8 Divisible by 4 or 6 = 25 + 16 - 8 = 33 P = 33/100 Answer: 33/100

Q23.

Two dice are thrown together. What is the probability that the sum is 10 or more?

  1. A1/12
  2. B5/36
  3. C1/6
  4. D1/3
Show answer & solution

Answer: C. 1/6

Solution

Sum 10: (4,6), (5,5), (6,4) = 3 Sum 11: (5,6), (6,5) = 2 Sum 12: (6,6) = 1 Favourable = 6; P = 6/36 = 1/6 Answer: 1/6

Q24.

A committee of 3 is chosen at random from 4 men and 3 women. What is the probability that all three are women?

  1. A3/35
  2. B1/7
  3. C4/35
  4. D1/35
Show answer & solution

Answer: D. 1/35

Solution

Total ways = 7C3 = 35 All women = 3C3 = 1 P = 1/35 Answer: 1/35

Q25.

Two dice are thrown. What is the probability that the product of the numbers is even?

  1. A3/4
  2. B1/4
  3. C1/2
  4. D5/6
Show answer & solution

Answer: A. 3/4

Solution

Product is odd only if both numbers are odd: 3 x 3 = 9 outcomes. P(odd) = 9/36 = 1/4 P(even) = 1 - 1/4 = 3/4 Answer: 3/4

Q26.

What is the probability that a randomly chosen leap year has 53 Sundays?

  1. A1/7
  2. B2/7
  3. C3/7
  4. D5/7
Show answer & solution

Answer: B. 2/7

Solution

A leap year has 366 days = 52 weeks + 2 extra days. The 2 extra days can be: (Mon,Tue), (Tue,Wed), (Wed,Thu), (Thu,Fri), (Fri,Sat), (Sat,Sun), (Sun,Mon) -> 7 cases. Sunday appears in 2 of them. P = 2/7 Answer: 2/7

Q27.

A bag has 3 red and 2 blue balls. Two balls are drawn one after another WITH replacement. What is the probability that they are of different colours?

  1. A3/5
  2. B6/25
  3. C12/25
  4. D3/10
Show answer & solution

Answer: C. 12/25

Solution

P(red then blue) = (3/5)(2/5) = 6/25 P(blue then red) = (2/5)(3/5) = 6/25 P = 6/25 + 6/25 = 12/25 Answer: 12/25

Q28.

A 3-digit number is chosen at random. What is the probability that it is divisible by 5?

  1. A1/10
  2. B2/9
  3. C179/900
  4. D1/5
Show answer & solution

Answer: D. 1/5

Solution

3-digit numbers: 100 to 999 = 900 Multiples of 5: 100, 105, ..., 995 = 180 P = 180/900 = 1/5 Answer: 1/5

Q29.

Three cards are drawn at random from a pack of 52 cards. What is the probability that all three are hearts?

  1. A11/850
  2. B1/64
  3. C1/32
  4. D11/425
Show answer & solution

Answer: A. 11/850

Solution

Total ways = 52C3 = 22,100 All hearts = 13C3 = 286 P = 286/22100 = 11/850 Answer: 11/850

Shortcut: (13/52)(12/51)(11/50) = 11/850

Q30.

A bag contains 5 red, 4 white and 3 black balls. Three balls are drawn at random. What is the probability that one ball of each colour is drawn?

  1. A1/11
  2. B3/11
  3. C2/11
  4. D5/11
Show answer & solution

Answer: B. 3/11

Solution

Total ways = 12C3 = 220 One of each colour = 5 x 4 x 3 = 60 P = 60/220 = 3/11 Answer: 3/11

Q31.

Two dice are thrown. What is the probability that the sum is a prime number?

  1. A1/3
  2. B7/18
  3. C5/12
  4. D11/36
Show answer & solution

Answer: C. 5/12

Solution

Prime sums: 2 (1 way), 3 (2), 5 (4), 7 (6), 11 (2) = 15 ways P = 15/36 = 5/12 Answer: 5/12

Q32.

A coin is tossed three times. What is the probability of getting at most two heads?

  1. A1/2
  2. B3/4
  3. C1/8
  4. D7/8
Show answer & solution

Answer: D. 7/8

Solution

P(3 heads) = 1/8 P(at most 2 heads) = 1 - 1/8 = 7/8 Answer: 7/8

Q33.

A card is drawn from a pack of 52 cards. What is the probability that it is a king or a heart?

  1. A4/13
  2. B17/52
  3. C1/4
  4. D2/13
Show answer & solution

Answer: A. 4/13

Solution

Kings = 4; hearts = 13; king of hearts is counted twice. Favourable = 4 + 13 - 1 = 16 P = 16/52 = 4/13 Answer: 4/13

Q34.

Two dice are thrown. What is the probability that at least one of them shows a 6?

  1. A1/3
  2. B1/6
  3. C5/36
  4. D11/36
Show answer & solution

Answer: D. 11/36

Solution

P(no 6 on either) = (5/6)(5/6) = 25/36 P(at least one 6) = 1 - 25/36 = 11/36 Answer: 11/36

Q35.

A bag has 5 red and 3 blue balls. Two balls are drawn one after the other without replacement. What is the probability that the first is red and the second is blue?

  1. A15/56
  2. B15/64
  3. C15/28
  4. D3/8
Show answer & solution

Answer: A. 15/56

Solution

P(first red) = 5/8 P(second blue | first red) = 3/7 (7 balls left, 3 blue) P = (5/8)(3/7) = 15/56 Answer: 15/56

Hard questions

Q36.

Machine A produces 60% of the items in a factory and machine B produces 40%. 2% of A's items and 5% of B's items are defective. An item is picked at random and found defective. What is the probability that it was made by A?

  1. A3/5
  2. B3/8
  3. C1/50
  4. D5/8
Show answer & solution

Answer: B. 3/8

Solution

P(A and defective) = 0.60 x 0.02 = 0.012 P(B and defective) = 0.40 x 0.05 = 0.020 P(defective) = 0.032 P(A | defective) = 0.012/0.032 = 3/8 Answer: 3/8

Q37.

Three shooters hit a target with probabilities 1/2, 1/3 and 1/4 respectively. If each fires once, what is the probability that exactly one of them hits the target?

  1. A1/24
  2. B13/24
  3. C11/24
  4. D3/4
Show answer & solution

Answer: C. 11/24

Solution

Exactly one hits: Only 1st: (1/2)(2/3)(3/4) = 6/24 Only 2nd: (1/2)(1/3)(3/4) = 3/24 Only 3rd: (1/2)(2/3)(1/4) = 2/24 Total = 11/24 Answer: 11/24

Q38.

Three fair dice are thrown together. What is the probability that the sum of the numbers is 10?

  1. A1/6
  2. B25/216
  3. C5/36
  4. D1/8
Show answer & solution

Answer: D. 1/8

Solution

Total outcomes = 6^3 = 216 Combinations for sum 10: (1,3,6)x6, (1,4,5)x6, (2,2,6)x3, (2,3,5)x6, (2,4,4)x3, (3,3,4)x3 = 27 ways P = 27/216 = 1/8 Answer: 1/8

Q39.

Numbers b and c are obtained from two throws of a fair die (b from the first, c from the second). What is the probability that the equation x^2 + bx + c = 0 has real roots?

  1. A19/36
  2. B17/36
  3. C1/2
  4. D5/9
Show answer & solution

Answer: A. 19/36

Solution

Real roots need b^2 >= 4c. b = 1: none; b = 2: c = 1 (1); b = 3: c = 1, 2 (2); b = 4: c = 1 to 4 (4); b = 5: c = 1 to 6 (6) as 25 >= 24; b = 6: c = 1 to 6 (6) Favourable = 0 + 1 + 2 + 4 + 6 + 6 = 19 P = 19/36 Answer: 19/36

Q40.

Urn I has 3 red and 2 blue balls; Urn II has 2 red and 4 blue balls. An urn is chosen at random and one ball is drawn from it. What is the probability that the ball is red?

  1. A1/2
  2. B7/15
  3. C5/11
  4. D8/15
Show answer & solution

Answer: B. 7/15

Solution

P(red) = P(I) x P(red|I) + P(II) x P(red|II) = (1/2)(3/5) + (1/2)(2/6) = 3/10 + 1/6 = 9/30 + 5/30 = 14/30 = 7/15 Answer: 7/15

Q41.

Urn I has 3 red and 2 blue balls; Urn II has 2 red and 4 blue balls. An urn is chosen at random and a ball is drawn; it turns out to be red. What is the probability that it came from Urn I?

  1. A3/5
  2. B5/14
  3. C9/14
  4. D1/2
Show answer & solution

Answer: C. 9/14

Solution

P(I and red) = (1/2)(3/5) = 3/10 P(red) = 3/10 + (1/2)(1/3) = 7/15 P(I | red) = (3/10)/(7/15) = (3/10)(15/7) = 9/14 Answer: 9/14

Q42.

Asha and Ravi toss a fair coin alternately, Asha first. The first person to get a head wins. What is the probability that Asha wins?

  1. A1/2
  2. B3/4
  3. C3/5
  4. D2/3
Show answer & solution

Answer: D. 2/3

Solution

Asha wins if the first head comes on toss 1, 3, 5, ... P = 1/2 + (1/2)^3 + (1/2)^5 + ... = (1/2)/(1 - 1/4) = 2/3 Answer: 2/3

Q43.

Three friends are each born on a day of the week, all days being equally likely and independent. What is the probability that all three were born on different days of the week?

  1. A30/49
  2. B6/49
  3. C1/7
  4. D5/7
Show answer & solution

Answer: A. 30/49

Solution

Total outcomes = 7^3 = 343 All different = 7 x 6 x 5 = 210 P = 210/343 = 30/49 Answer: 30/49

Q44.

Two different numbers are chosen at random from 1 to 50. What is the probability that their sum is even?

  1. A1/2
  2. B24/49
  3. C25/49
  4. D23/49
Show answer & solution

Answer: B. 24/49

Solution

Sum is even if both are even or both are odd. Both even: 25C2 = 300; both odd: 25C2 = 300 Total pairs = 50C2 = 1225 P = 600/1225 = 24/49 Answer: 24/49

Q45.

Five cards are drawn at random from a pack of 52 cards. What is the probability that all five are of the same suit?

  1. A1/1024
  2. B33/66640
  3. C33/16660
  4. D1/256
Show answer & solution

Answer: C. 33/16660

Solution

Choose the suit: 4 ways; choose 5 cards from its 13: 13C5 = 1287 Favourable = 4 x 1287 = 5148 Total = 52C5 = 2,598,960 P = 5148/2598960 = 33/16660 Answer: 33/16660

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