Probability

What to remember

• Classical probability: P(E) = (number of outcomes favourable to E) / (total number of equally likely outcomes). Always 0 ≤ P(E) ≤ 1.
• Complementary events: P(E) + P(not E) = 1. A sure event has probability 1 and an impossible event has probability 0.
• Sample space is the set of all possible outcomes. The sum of the probabilities of all elementary events of an experiment is 1.

Basic terms

• Experiment (trial): an action whose result is not known in advance, such as tossing a coin or rolling a die.
• Outcome: one possible result of the trial.
• Sample space (S): the set of all outcomes.
• Event (E): a subset of the sample space. An elementary event has only one outcome.
• Equally likely outcomes: outcomes with the same chance. A fair coin gives head or tail with equal chance. A fair die gives each face with equal chance.
• Sure (certain) event: it always happens. P = 1. Example: getting a number from 1 to 6 on a die.
• Impossible event: it can never happen. P = 0. Example: getting 7 on a single die.
• Complement of E (written E′ or "not E"): all outcomes of S that are not in E.
• Mutually exclusive events: they cannot happen together. Getting a head and getting a tail in one toss are mutually exclusive.

Two kinds of probability

Experimental (empirical) probability. Found from actual trials:

P(E) = (number of trials in which E happened) / (total number of trials)

If a coin is tossed 100 times and heads appear 48 times, the experimental probability of a head is 48/100 = 0.48. When the number of trials grows large, this value comes close to the theoretical value 0.5.

Theoretical (classical) probability. Found by reasoning, without doing the trials:

P(E) = n(E) / n(S)

This works only when all outcomes are equally likely.

Probability can be written as a fraction, a decimal or a percentage. Probability can never be negative and can never be more than 1.

Standard sample spaces

A rule: n coins give 2ⁿ outcomes; n dice give 6ⁿ outcomes.

Playing cards

A standard pack has 52 cards in 4 suits of 13 cards each: spades and clubs (black), hearts and diamonds (red). So there are 26 red and 26 black cards. Each suit has an ace, 2 to 10, a jack, a queen and a king. There are 4 aces, 4 kings, 4 queens and 4 jacks. Face cards are the jack, queen and king: 3 per suit, so 12 in all.

Worked examples

1. Single die. P(getting a prime number) = outcomes 2, 3, 5 = 3/6 = 1/2. P(number greater than 4) = outcomes 5, 6 = 2/6 = 1/3. P(even number) = 3/6 = 1/2.

2. Two coins. P(two heads) = 1/4. P(at least one head) = 3/4 (HH, HT, TH). P(exactly one head) = 2/4 = 1/2. P(no head) = 1/4.

3. Three coins. P(exactly two heads) = 3/8 (HHT, HTH, THH). P(at least one head) = 1 − P(no head) = 1 − 1/8 = 7/8.

4. Two dice. Total outcomes = 36.

The sum 7 is the most likely sum with two dice. The sums 2 and 12 are the least likely.

5. Bag of balls. A bag has 5 red, 3 blue and 2 green balls. Total = 10. P(blue) = 3/10. P(not red) = 1 − 5/10 = 1/2. P(red or green) = 7/10.

6. Cards. P(a king) = 4/52 = 1/13. P(a red card) = 26/52 = 1/2. P(a face card) = 12/52 = 3/13. P(a red face card) = 6/52 = 3/26. P(the ace of hearts) = 1/52. P(a spade) = 13/52 = 1/4.

7. Numbers 1 to 25. P(multiple of 3 or 5): multiples of 3 are 8 (3 to 24), multiples of 5 are 5, and 15 is common. Count = 8 + 5 − 1 = 12. Probability = 12/25.

8. Numbers 1 to 20. Primes are 2, 3, 5, 7, 11, 13, 17, 19, which is 8. P(prime) = 8/20 = 2/5.

9. Calendar. A non-leap year has 365 days = 52 weeks + 1 day. The extra day can be any of 7 weekdays, so P(53 Sundays) = 1/7. A leap year has 52 weeks + 2 days, so P(53 Sundays) = 2/7.

10. Complement. If P(E) = 0.35, then P(not E) = 0.65. If the chance of winning is 3/8, the chance of not winning is 5/8.

11. Unknown count. A bag has 12 balls, some are black. If P(black) = 1/3, the number of black balls = 12 × 1/3 = 4. If 6 more black balls are added, P(black) = 10/18 = 5/9.

Odds and fair games

If P(E) = a/b, the odds in favour of E are a : (b − a). For a die, the odds in favour of a 6 are 1 : 5. A game is fair when each player has the same chance of winning, such as deciding the first turn by tossing a fair coin.

Useful counting ideas

• Union of events: n(A or B) = n(A) + n(B) − n(A and B). Divide by n(S) to get the probability. For mutually exclusive events the last term is zero, so P(A or B) = P(A) + P(B).
• Without replacement: if one card is drawn and not put back, the pack has 51 cards for the next draw. Class 10 problems usually draw only one item at a time, so this is rare.
• Arranging digits: from the digits 1, 2, 3 the number of two-digit numbers with different digits is 6 (12, 13, 21, 23, 31, 32). The probability of an even one is 2/6 = 1/3 (12 and 32).
• Quick check: after finding all probabilities in a problem, add them. If the full list of elementary events does not sum to 1, a case is missing.

Classroom angle

• Probability is learned best through activities. Let students toss coins, roll dice and draw cards from a cap, and record results in a table. Pool the class results to show that experimental probability moves close to theoretical probability as trials increase.
• Clear the common mistake: "after 5 heads in a row a tail is due". Each toss is independent. The coin has no memory.
• Use a spinner or a bag of coloured counters for equal and unequal chances. Show that "equally likely" must be checked before using the classical formula. A drawing pin may land point up or point down, but these two outcomes are not equally likely.
• Ask students to write sentences with words such as "certain", "likely", "unlikely", "impossible" and place them on a 0 to 1 line.

Exam traps

• Probability of an event is never more than 1 and never negative. An answer like 7/5 is wrong.
• Two dice have 36 outcomes, not 12. (1,2) and (2,1) are different outcomes.
• "At least one" is best found as 1 − P(none).
• Face cards are 12, not 16; the ace is not counted as a face card in school mathematics.
• A probability question may ask for "red or a king". Do not count the two red kings twice: 26 + 4 − 2 = 28 cards, so the probability is 28/52 = 7/13.
• Selecting a "letter of the word" needs counting letters, not distinct letters. In MATHEMATICS there are 11 letters, and M appears twice, so P(M) = 2/11.
• In 1 to 20, the number 1 is not prime, and 2 is the only even prime.
• Experimental probability changes from one set of trials to another; theoretical probability is fixed.

One-liners

• 1. P(sure event) = 1 and P(impossible event) = 0.
• 2. 0 ≤ P(E) ≤ 1.
• 3. P(not E) = 1 − P(E).
• 4. A fair die has 6 equally likely outcomes.
• 5. Two dice give 36 outcomes; three coins give 8.
• 6. A pack of cards has 52 cards, 26 red and 26 black.
• 7. There are 12 face cards and 4 aces in a pack.
• 8. P(sum 7 with two dice) = 1/6.
• 9. P(a doublet with two dice) = 1/6.
• 10. A non-leap year gives 53 Sundays with probability 1/7.
• 11. Sum of probabilities of all elementary events = 1.
• 12. Experimental probability approaches theoretical probability as trials increase.

Practice questions

1. The probability of a sure event is

   1. 1
   2. 0
   3. 1/2
   4. 100
   Answer

   A. 1

   A sure event always occurs, so P = 1.

2. The probability of an impossible event is

   1. 1
   2. 0
   3. -1
   4. 1/2
   Answer

   B. 0

   An impossible event never occurs, so P = 0.

3. Which of the following cannot be the probability of an event?

   1. 0
   2. 0.45
   3. 7/5
   4. 1
   Answer

   C. 7/5

   Probability lies between 0 and 1; 7/5 is greater than 1.

4. If P(E) is the probability of an event E, then P(not E) is

   1. 1 + P(E)
   2. P(E)/2
   3. P(E) - 1
   4. 1 - P(E)
   Answer

   D. 1 - P(E)

   E and not E together make the whole sample space.

5. The number of possible outcomes when two coins are tossed together is

   1. 4
   2. 2
   3. 3
   4. 6
   Answer

   A. 4

   HH, HT, TH, TT.

6. The number of possible outcomes when two dice are thrown together is

   1. 6
   2. 36
   3. 18
   4. 12
   Answer

   B. 36

   6 × 6 = 36 ordered pairs.

7. The number of cards in a standard pack, without jokers, is

   1. 48
   2. 56
   3. 54
   4. 52
   Answer

   D. 52

   4 suits of 13 cards each.

8. The number of face cards in a standard pack is

   1. 16
   2. 4
   3. 13
   4. 12
   Answer

   D. 12

   Jack, queen and king in each of the 4 suits: 3 × 4 = 12.

9. The number of red cards in a pack of 52 cards is

   1. 13
   2. 12
   3. 26
   4. 52
   Answer

   C. 26

   Hearts and diamonds have 13 cards each.

10. The most likely sum when two dice are thrown is

    1. 6
    2. 7
    3. 12
    4. 2
    Answer

    B. 7

    Sum 7 can occur in 6 ways, more than any other sum.

11. Experimental probability of an event is

    1. number of outcomes divided by number of events
    2. number of trials in which it occurs divided by the total number of trials
    3. always equal to 1/2
    4. fixed for all sets of trials
    Answer

    B. number of trials in which it occurs divided by the total number of trials

    It is found by actually performing trials and recording results.

12. The probability of getting a number greater than 6 when a die is rolled once is

    1. 1/6
    2. 1
    3. 1/3
    4. 0
    Answer

    D. 0

    No face shows a number above 6, so the event is impossible.

13. The probability of getting an even number when a die is rolled once is

    1. 1/2
    2. 1/3
    3. 2/3
    4. 1/6
    Answer

    A. 1/2

    Favourable outcomes 2, 4, 6 out of 6.

14. The probability of getting a doublet when two dice are thrown is

    1. 1/12
    2. 1/3
    3. 1/6
    4. 1/36
    Answer

    C. 1/6

    Six doublets (1,1) to (6,6) out of 36.

15. The sum of the probabilities of all elementary events of an experiment is

    1. 0
    2. 2
    3. 1
    4. n
    Answer

    C. 1

    All outcomes together make the sample space with probability 1.

16. Outcomes are said to be equally likely when

    1. they are mutually exclusive
    2. each has the same chance of occurring
    3. the sum is 1
    4. each has probability 1
    Answer

    B. each has the same chance of occurring

    Equal chance is the meaning of equally likely.

17. The number of aces in a standard pack is

    1. 4
    2. 1
    3. 2
    4. 13
    Answer

    A. 4

    One ace in each suit.

18. A die is rolled once. The probability of getting a prime number is

    1. 1/6
    2. 2/3
    3. 1/3
    4. 1/2
    Answer

    D. 1/2

    Primes are 2, 3, 5: 3 out of 6.

19. Two coins are tossed. The probability of getting at least one head is

    1. 1/4
    2. 1/2
    3. 3/4
    4. 1
    Answer

    C. 3/4

    Favourable HH, HT, TH out of 4.

20. Three coins are tossed. The probability of getting exactly two heads is

    1. 1/2
    2. 1/8
    3. 1/4
    4. 3/8
    Answer

    D. 3/8

    HHT, HTH, THH are favourable out of 8.

21. A bag has 5 red, 3 blue and 2 green balls. The probability of drawing a blue ball is

    1. 3/10
    2. 1/3
    3. 3/7
    4. 3/5
    Answer

    A. 3/10

    3 blue out of 10 balls.

22. A bag has 5 red, 3 blue and 2 green balls. The probability that the ball drawn is not red is

    1. 2/5
    2. 1/2
    3. 1/5
    4. 3/10
    Answer

    B. 1/2

    1 - 5/10 = 1/2.

23. One card is drawn from a well-shuffled pack of 52 cards. The probability that it is a king is

    1. 1/13
    2. 4/13
    3. 1/4
    4. 1/52
    Answer

    A. 1/13

    4 kings out of 52 cards.

24. One card is drawn from a well-shuffled pack. The probability that it is a red face card is

    1. 3/13
    2. 3/26
    3. 1/13
    4. 6/13
    Answer

    B. 3/26

    Red face cards are 6 out of 52: 6/52 = 3/26.

25. A number is chosen at random from 1 to 25. The probability that it is a multiple of 3 or 5 is

    1. 13/25
    2. 3/5
    3. 8/25
    4. 12/25
    Answer

    D. 12/25

    Multiples of 3: 8; of 5: 5; of 15: 1. So 8 + 5 - 1 = 12 favourable.

26. A number is chosen at random from 1 to 20. The probability that it is prime is

    1. 9/20
    2. 3/10
    3. 2/5
    4. 1/2
    Answer

    C. 2/5

    Primes: 2, 3, 5, 7, 11, 13, 17, 19, which is 8 numbers: 8/20 = 2/5.

27. The probability that a non-leap year has 53 Sundays is

    1. 1/52
    2. 1/7
    3. 2/7
    4. 53/365
    Answer

    B. 1/7

    365 days = 52 weeks + 1 extra day; that day must be Sunday: 1 of 7.

28. If P(E) = 0.35, then P(not E) is

    1. 0.65
    2. 0.35
    3. 1.35
    4. 0.55
    Answer

    A. 0.65

    1 - 0.35 = 0.65.

29. A bag has 12 balls and the probability of drawing a black ball is 1/3. The number of black balls is

    1. 4
    2. 3
    3. 6
    4. 9
    Answer

    A. 4

    12 × 1/3 = 4.

30. A coin is tossed 100 times and heads appear 48 times. The experimental probability of a head is

    1. 48
    2. 0.5
    3. 0.48
    4. 0.52
    Answer

    C. 0.48

    48/100 = 0.48.

31. Two dice are thrown. The probability that the sum is 7 is

    1. 5/36
    2. 7/36
    3. 1/12
    4. 1/6
    Answer

    D. 1/6

    Six pairs give sum 7 out of 36.

32. Two dice are thrown. The probability that the sum is less than 4 is

    1. 1/9
    2. 1/18
    3. 1/12
    4. 1/6
    Answer

    C. 1/12

    Sums 2 and 3: (1,1), (1,2), (2,1) which is 3 of 36.

33. A letter is chosen at random from the word MATHEMATICS. The probability that it is A is

    1. 2/5
    2. 1/11
    3. 1/5
    4. 2/11
    Answer

    D. 2/11

    The word has 11 letters and A appears twice.

34. A bag has 12 balls and 4 are black. If 6 more black balls are added, the probability of drawing a black ball is

    1. 2/3
    2. 5/9
    3. 1/3
    4. 10/12
    Answer

    B. 5/9

    Black = 10 out of 18 balls: 10/18 = 5/9.

35. A number is formed from digits 1, 2, 3 using two different digits. The probability that it is even is

    1. 1/6
    2. 1/2
    3. 2/3
    4. 1/3
    Answer

    D. 1/3

    Numbers: 12, 13, 21, 23, 31, 32. Even ones are 12 and 32: 2/6 = 1/3.

36. One card is drawn from a pack of 52 cards. The probability that it is a red card or a king is

    1. 7/13
    2. 8/13
    3. 4/13
    4. 1/2
    Answer

    A. 7/13

    Red cards 26 plus black kings 2 = 28 cards; 28/52 = 7/13.

37. Consider the statements. 1. The probability of an event lies between 0 and 1, both included. 2. The probability of an event can be negative. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Probability is never negative.

38. Consider the statements. 1. Two dice thrown together have 36 outcomes. 2. The outcomes (1, 2) and (2, 1) are the same. Which is/are correct?

    1. 2 only
    2. 1 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 1 only

    (1,2) and (2,1) are different ordered outcomes.

39. Consider the statements. 1. The probability of a sure event is 0. 2. The probability of an impossible event is 0. Which is/are correct?

    1. 1 only
    2. Both 1 and 2
    3. 2 only
    4. Neither 1 nor 2
    Answer

    C. 2 only

    A sure event has probability 1.

40. Consider the statements. 1. Experimental probability is found by performing trials. 2. Classical probability formula is used when outcomes are equally likely. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Neither 1 nor 2
    4. Both 1 and 2
    Answer

    D. Both 1 and 2

    Both describe the two kinds of probability correctly.

41. Consider the statements. 1. After five heads in a row, a tail becomes more likely on the next toss of a fair coin. 2. Each toss of a fair coin is independent of earlier tosses. Which is/are correct?

    1. 2 only
    2. 1 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 2 only

    A coin has no memory; the chance of a tail stays 1/2.

42. Consider the statements. 1. If P(E) = 0.2 then P(not E) = 0.8. 2. If the chance of winning is 3/8 then the chance of not winning is 5/8. Which is/are correct?

    1. 1 only
    2. Both 1 and 2
    3. 2 only
    4. Neither 1 nor 2
    Answer

    B. Both 1 and 2

    Complementary probabilities add to 1.

43. Consider the statements. 1. The probability of a sum of 13 with two dice is 1/36. 2. The probability of a sum of 7 with two dice is 1/12. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Neither 1 nor 2
    4. Both 1 and 2
    Answer

    C. Neither 1 nor 2

    A sum of 13 is impossible (probability 0) and sum 7 has probability 6/36 = 1/6.

44. Match the event (one die) with its probability. P. Getting a number at most 6 Q. Getting 7 R. Getting an even number 1. 0 2. 1/2 3. 1

    1. P-3, Q-2, R-1
    2. P-2, Q-1, R-3
    3. P-1, Q-3, R-2
    4. P-3, Q-1, R-2
    Answer

    D. P-3, Q-1, R-2

    The sure event has probability 1, the impossible event 0, and an even number 3/6 = 1/2.

45. Match the event (one card from a pack) with its probability. P. A spade Q. A king R. A red card 1. 1/13 2. 1/2 3. 1/4

    1. P-1, Q-3, R-2
    2. P-2, Q-1, R-3
    3. P-3, Q-1, R-2
    4. P-3, Q-2, R-1
    Answer

    C. P-3, Q-1, R-2

    Spades 13/52 = 1/4; kings 4/52 = 1/13; red cards 26/52 = 1/2.