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Probability aptitude questions

Favourable over total, complements, and independent events.

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Formula sheetProbability: the essentialsFormula sheetProbability: the essentialsShow

  • P(E) = favourable outcomes ÷ total outcomes, always between 0 and 1, and P(not E) = 1 − P(E).
  • Or: P(A or B) = P(A) + P(B) − P(A and B). Drop the last term only for mutually exclusive events.
  • And, for independent events: P(A and B) = P(A) × P(B).
  • At least one = 1 − P(none). A problem given to independent solvers is solved with probability 1 − the product of their failure probabilities.
  • Without replacement, reduce both the favourable count and the total after each draw, or count with combinations: two reds from 4 red and 6 blue balls is C(4, 2)/C(10, 2) = 6/45.
  • Coins: n coins give 2ⁿ outcomes, and exactly k heads has probability C(n, k)/2ⁿ.
  • Two dice: 36 outcomes. A sum s can be made in s − 1 ways for s ≤ 7 and 13 − s ways for s ≥ 7; there are 6 doublets.
  • Cards: 52 cards, 4 suits of 13, 26 red and 26 black, 12 face cards (jack, queen, king) and 4 aces.
  • Calendars: a leap year has 53 of a given weekday with probability 2/7, an ordinary year 1/7.
  • Numbers from 1 to n divisible by d: ⌊n/d⌋. For '3 or 5', subtract the multiples of 15.

Example: a red card or a king is 26/52 + 4/52 − 2/52 = 28/52 = 7/13.

Questions37

1

Sum of two dice is 9

Two fair dice are rolled. What is the probability that the sum is 9?

dice

45s

Easy2

A king or a heart

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

cards · union

45s

Easy3

Two reds without replacement

A bag has 4 red and 6 blue balls. Two balls are drawn one after another without replacement. What is the probability that both are red?

without replacement

45s

Medium4

At least two heads in three tosses

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

coins

45s

Easy5

53 Sundays in a leap year

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

calendar

60s

Medium6

Solved by at least one of them

A can solve a problem with probability 2/3 and B with probability 3/4, independently. What is the probability that the problem is solved?

independent events

45s

Medium7

Sum of two dice is 8

Two fair dice are thrown. What is the probability that the sum is 8?

dice

60s

Easy8

Throwing a doublet

Two dice are thrown together. What is the probability of getting a doublet?

dice

45s

Easy9

Drawing a face card

One card is drawn from a well-shuffled pack of 52. What is the probability that it is a face card?

cards

45s

Easy10

A queen or a club

A card is drawn from a pack of 52. What is the probability that it is a queen or a club?

cards · union

60s

Medium11

Two kings in a row

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

cards · without replacement

75s

Medium12

Two red balls

A bag contains 5 red and 4 green balls. Two are drawn at random. What is the probability that both are red?

balls

75s

Medium13

One of each colour

A bag has 4 white and 5 black balls. Two are drawn at random. What is the probability that they are of different colours?

balls

75s

Medium14

Exactly two heads

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

coins

45s

Easy15

No heads at all

Four fair coins are tossed. What is the probability of getting no heads?

coins

45s

Easy16

53 Mondays in a leap year

What is the probability that a leap year chosen at random has 53 Mondays?

calendar

75s

Medium17

53 Sundays in an ordinary year

What is the probability that a non-leap year has 53 Sundays?

calendar

60s

Medium18

At least one solves it

A can solve a problem with probability 1/2 and B with probability 1/3, independently. What is the probability that the problem gets solved?

independent events

75s

Medium19

Both hit the target

Two marksmen hit a target with probabilities 3/5 and 2/3. What is the probability that both hit it?

independent events

45s

Easy20

Divisible by 5

A number is chosen at random from 1 to 50. What is the probability that it is divisible by 5?

numbers

45s

Easy21

Picking a prime

A number is picked at random from 1 to 20. What is the probability that it is prime?

numbers

45s

Easy22

A multiple of 3 or 5

Tickets numbered 1 to 20 are mixed and one is drawn. What is the probability that it is a multiple of 3 or 5?

numbers · union

75s

Medium23

Both children girls

A family has two children. What is the probability that both are girls, assuming each child is equally likely to be a boy or a girl?

basics

45s

Easy24

Drawing a defective bulb

A box has 20 bulbs, of which 4 are defective. One bulb is drawn at random. What is the probability that it is not defective?

basics

45s

Easy25

Two good bulbs

From a box of 10 bulbs of which 3 are defective, two are drawn at random. What is the probability that both are good?

balls

75s

Medium26

Sum greater than nine

Two dice are thrown. What is the probability that the sum is greater than 9?

dice

75s

Medium27

A red king

One card is drawn from a pack of 52. What is the probability that it is a red king?

cards

45s

Easy28

Two balls of the same colour

A bag holds 3 red and 5 blue balls. Two are drawn at random. What is the probability that both are the same colour?

balls

90s

Hard29

Exactly three heads

Four fair coins are tossed. What is the probability of getting exactly three heads?

coins

60s

Medium30

At least one head

Three fair coins are tossed. What is the probability of getting at least one head?

coins

45s

Easy31

Divisible by 3 or 5

A number is chosen at random from 1 to 30. What is the probability that it is divisible by 3 or 5?

numbers · union

90s

Medium32

Selecting two students

From 4 boys and 6 girls, two students are chosen at random. What is the probability that both are girls?

balls

75s

Medium33

Neither passes

Two students have probabilities 0.6 and 0.7 of passing an exam, independently. What is the probability that neither passes?

independent events

60s

Medium34

A ticket ending in a given digit

Tickets numbered 1 to 40 are placed in a box and one is drawn. What is the probability that it is a multiple of 4?

numbers

45s

Easy35

Both dice even

Two dice are thrown. What is the probability that both show even numbers?

dice

45s

Easy36

Not a face card

A card is drawn from a pack of 52. What is the probability that it is not a face card?

cards

45s

Easy37

Either one or the other

P(A) = 0.4, P(B) = 0.5, and P(A and B) = 0.2. What is P(A or B)?

union

45s

Easy

Showing 37 of 37

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Probability Aptitude Questions with Solutions

37 Probability questions with hints, worked solutions and time targets — 19 easy · 17 medium · 1 hard.

  • Section: Quantitative Aptitude
  • Questions: 37
  • By difficulty: 19 easy · 17 medium · 1 hard
  • Cost: Free, unlimited on every plan

Favourable over total, complements, and independent events.

Probability: the essentials

  • P(E) = favourable outcomes ÷ total outcomes, always between 0 and 1, and P(not E) = 1 − P(E).
  • Or: P(A or B) = P(A) + P(B) − P(A and B). Drop the last term only for mutually exclusive events.
  • And, for independent events: P(A and B) = P(A) × P(B).
  • At least one = 1 − P(none). A problem given to independent solvers is solved with probability 1 − the product of their failure probabilities.
  • Without replacement, reduce both the favourable count and the total after each draw, or count with combinations: two reds from 4 red and 6 blue balls is C(4, 2)/C(10, 2) = 6/45.
  • Coins: n coins give 2ⁿ outcomes, and exactly k heads has probability C(n, k)/2ⁿ.
  • Two dice: 36 outcomes. A sum s can be made in s − 1 ways for s ≤ 7 and 13 − s ways for s ≥ 7; there are 6 doublets.
  • Cards: 52 cards, 4 suits of 13, 26 red and 26 black, 12 face cards (jack, queen, king) and 4 aces.
  • Calendars: a leap year has 53 of a given weekday with probability 2/7, an ordinary year 1/7.
  • Numbers from 1 to n divisible by d: ⌊n/d⌋. For '3 or 5', subtract the multiples of 15.

Example: a red card or a king is 26/52 + 4/52 − 2/52 = 28/52 = 7/13.

Questions

  1. Sum of two dice is 9 Easy
    Two fair dice are rolled. What is the probability that the sum is 9?
  2. A king or a heart Easy
    One card is drawn from a standard pack of 52. What is the probability that it is a king or a heart?
  3. Two reds without replacement Medium
    A bag has 4 red and 6 blue balls. Two balls are drawn one after another without replacement. What is the probability that both are red?
  4. At least two heads in three tosses Easy
    Three fair coins are tossed. What is the probability of getting at least two heads?
  5. 53 Sundays in a leap year Medium
    What is the probability that a leap year has 53 Sundays?
  6. Solved by at least one of them Medium
    A can solve a problem with probability 2/3 and B with probability 3/4, independently. What is the probability that the problem is solved?
  7. Sum of two dice is 8 Easy
    Two fair dice are thrown. What is the probability that the sum is 8?
  8. Throwing a doublet Easy
    Two dice are thrown together. What is the probability of getting a doublet?
  9. Drawing a face card Easy
    One card is drawn from a well-shuffled pack of 52. What is the probability that it is a face card?
  10. A queen or a club Medium
    A card is drawn from a pack of 52. What is the probability that it is a queen or a club?
  11. Two kings in a row Medium
    Two cards are drawn without replacement from a pack of 52. What is the probability that both are kings?
  12. Two red balls Medium
    A bag contains 5 red and 4 green balls. Two are drawn at random. What is the probability that both are red?
  13. One of each colour Medium
    A bag has 4 white and 5 black balls. Two are drawn at random. What is the probability that they are of different colours?
  14. Exactly two heads Easy
    Three fair coins are tossed. What is the probability of getting exactly two heads?
  15. No heads at all Easy
    Four fair coins are tossed. What is the probability of getting no heads?
  16. 53 Mondays in a leap year Medium
    What is the probability that a leap year chosen at random has 53 Mondays?
  17. 53 Sundays in an ordinary year Medium
    What is the probability that a non-leap year has 53 Sundays?
  18. At least one solves it Medium
    A can solve a problem with probability 1/2 and B with probability 1/3, independently. What is the probability that the problem gets solved?
  19. Both hit the target Easy
    Two marksmen hit a target with probabilities 3/5 and 2/3. What is the probability that both hit it?
  20. Divisible by 5 Easy
    A number is chosen at random from 1 to 50. What is the probability that it is divisible by 5?
  21. Picking a prime Easy
    A number is picked at random from 1 to 20. What is the probability that it is prime?
  22. A multiple of 3 or 5 Medium
    Tickets numbered 1 to 20 are mixed and one is drawn. What is the probability that it is a multiple of 3 or 5?
  23. Both children girls Easy
    A family has two children. What is the probability that both are girls, assuming each child is equally likely to be a boy or a girl?
  24. Drawing a defective bulb Easy
    A box has 20 bulbs, of which 4 are defective. One bulb is drawn at random. What is the probability that it is not defective?
  25. Two good bulbs Medium
    From a box of 10 bulbs of which 3 are defective, two are drawn at random. What is the probability that both are good?
  26. Sum greater than nine Medium
    Two dice are thrown. What is the probability that the sum is greater than 9?
  27. A red king Easy
    One card is drawn from a pack of 52. What is the probability that it is a red king?
  28. Two balls of the same colour Hard
    A bag holds 3 red and 5 blue balls. Two are drawn at random. What is the probability that both are the same colour?
  29. Exactly three heads Medium
    Four fair coins are tossed. What is the probability of getting exactly three heads?
  30. At least one head Easy
    Three fair coins are tossed. What is the probability of getting at least one head?
  31. Divisible by 3 or 5 Medium
    A number is chosen at random from 1 to 30. What is the probability that it is divisible by 3 or 5?
  32. Selecting two students Medium
    From 4 boys and 6 girls, two students are chosen at random. What is the probability that both are girls?
  33. Neither passes Medium
    Two students have probabilities 0.6 and 0.7 of passing an exam, independently. What is the probability that neither passes?
  34. A ticket ending in a given digit Easy
    Tickets numbered 1 to 40 are placed in a box and one is drawn. What is the probability that it is a multiple of 4?
  35. Both dice even Easy
    Two dice are thrown. What is the probability that both show even numbers?
  36. Not a face card Easy
    A card is drawn from a pack of 52. What is the probability that it is not a face card?
  37. Either one or the other Easy
    P(A) = 0.4, P(B) = 0.5, and P(A and B) = 0.2. What is P(A or B)?

Placement tests that ask probability