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Sample space · 3 marks

For the following experiments write down the sample space S.

  1. (i) Rolling a die and tossing a coin together.
  2. (ii) Choosing a random integer between – 5 and + 5.
  3. (iii) A box containing 5 green and 7 red balls. One ball is drawn at random.
Answer: (i) S = {1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T}, 12 outcomes. (ii) S = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5}, 11 outcomes. (iii) S = {G1, G2, G3, G4, G5, R1, R2, …, R7}, 12 outcomes (by colour only: {Green, Red}).

Step-by-step solution

Idea: List every possible outcome once. When two things happen together, each outcome is a pair. When objects look alike, give each a label so that every object is a separate outcome.

(i) Rolling a die and tossing a coin together.

  1. Each outcome is a pair: (die number, coin face). Each of the 6 numbers can go with H or T, so there are 6 × 2 = 12 outcomes.½ mark
  2. S = {1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T}.½ mark
S = {1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T}; n(S) = 12

(ii) Choosing a random integer between – 5 and + 5.

  1. The integers from −5 to +5 are the negative integers −5 to −1, zero, and the positive integers 1 to 5. Do not forget 0, which is an integer.½ mark
  2. S = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5}, n(S) = 11. (If “between” is read as leaving out −5 and +5 themselves, S = {−4, −3, −2, −1, 0, 1, 2, 3, 4} with 9 outcomes. State which reading you use.)½ mark
S = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5}; n(S) = 11

(iii) A box containing 5 green and 7 red balls. One ball is drawn at random.

  1. Any of the 12 balls can be drawn. Label the green balls G1 to G5 and the red balls R1 to R7 so that each ball is a separate, equally likely outcome.½ mark
  2. S = {G1, G2, G3, G4, G5, R1, R2, R3, R4, R5, R6, R7}, n(S) = 12. If we record only the colour, S = {Green, Red}, but these two outcomes are not equally likely (512 and 712).½ mark
S = {G1, …, G5, R1, …, R7}; n(S) = 12 (colour only: {Green, Red})
(i) {1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T} (12 outcomes). (ii) {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5} (11 outcomes). (iii) {G1, G2, G3, G4, G5, R1, R2, R3, R4, R5, R6, R7} (12 outcomes); recording colour only gives {Green, Red}.

Check: (i) 6 die faces × 2 coin faces = 12 ✓. (ii) 5 negatives + zero + 5 positives = 11 ✓. (iii) 5 + 7 = 12 balls ✓.

Answer to write in the exam

(i)

Outcomes = 6 × 2 = 12

∴ S = {1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T}

(ii)

∴ S = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5}; n(S) = 11

(iii)

Green balls G1–G5, red balls R1–R7

∴ S = {G1, G2, G3, G4, G5, R1, R2, R3, R4, R5, R6, R7}; n(S) = 12

Common mistakes that cost marks

  • In (i), writing {1, 2, 3, 4, 5, 6, H, T}. The die and coin happen together, so each outcome is a pair such as 3H.
  • In (ii), leaving out 0. Zero is an integer.
  • In (iii), writing {Green, Red} and then saying each has probability 12. There are more red balls, so red is more likely (712).

How this can come in the exam

MCQ (1 mark)

A coin is tossed and a spinner with sectors A, B, C is spun together. The number of outcomes in the sample space is

  1. 3
  2. 5
  3. 6
  4. 8
Show answer

(C) 6
2 coin faces × 3 sectors = 6: {HA, HB, HC, TA, TB, TC}.

Short answer (2 marks)

Write the sample space for choosing a random whole number from 0 to 6, and for choosing a random even number from 1 to 9. State n(S) for each.

Show answer{0, 1, 2, 3, 4, 5, 6}: n(S) = 7. {2, 4, 6, 8}: n(S) = 4.

Try one yourself

A bag has 2 white and 3 black counters. One counter is drawn. Write the sample space with each counter labelled.

Show answer

S = {W1, W2, B1, B2, B3}, n(S) = 5.

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