Learnify Academy is a tuition centre in Bahrain. Classes are for students in Bahrain only.Tuition classes in Bahrain only

Tree diagrams · 4 marks

A box contains 4 balls numbered 1 to 4. Record a sample space using a tree diagram for the following experiments:

  1. (i) A ball is drawn, and the number is recorded. Then the ball is returned, and a second ball is drawn and recorded.
  2. (ii) A ball is drawn and recorded. Without replacing the first ball, the experimenter draws and records a second ball.
  3. (iii) What are the sizes of these two sample spaces?
Answer: (i) With replacement: 4 × 4 = 16 outcomes, (1, 1) to (4, 4). (ii) Without replacement: 4 × 3 = 12 outcomes (no repeated number). (iii) n(S) = 16 and 12.

Step-by-step solution

Idea: Each outcome is an ordered pair (first number, second number). If the ball is returned, the second draw has all 4 numbers again; if not, the first number cannot come up again, so each branch has only 3 second draws.

(i) with replacement11(1, 1)2(1, 2)3(1, 3)4(1, 4)21(2, 1)2(2, 2)3(2, 3)4(2, 4)31(3, 1)2(3, 2)3(3, 3)4(3, 4)41(4, 1)2(4, 2)3(4, 3)4(4, 4)(ii) no replacement12(1, 2)3(1, 3)4(1, 4)21(2, 1)3(2, 3)4(2, 4)31(3, 1)2(3, 2)4(3, 4)41(4, 1)2(4, 2)3(4, 3)

(i) A ball is drawn, and the number is recorded. Then the ball is returned, and a second ball is drawn and recorded.

  1. Tree: 4 branches for the first draw (1, 2, 3, 4). The ball is returned, so from each of them draw 4 branches (1, 2, 3, 4) for the second draw (left tree).½ mark
  2. S = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4)}.1 mark
16 ordered pairs, (1, 1) to (4, 4)

(ii) A ball is drawn and recorded. Without replacing the first ball, the experimenter draws and records a second ball.

  1. Tree: 4 branches for the first draw. The first ball is not replaced, so from each of them there are only 3 branches: the other three numbers (right tree).½ mark
  2. S = {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)}. Pairs such as (1, 1) are impossible now.1 mark
12 ordered pairs, no repeated number

(iii) What are the sizes of these two sample spaces?

  1. (i) n(S) = 4 × 4 = 16. (ii) n(S) = 4 × 3 = 12. The difference is the 4 pairs (1, 1), (2, 2), (3, 3), (4, 4), which can only happen with replacement.1 mark
16 and 12
(i) With replacement: S = {(1, 1), (1, 2), …, (4, 4)}, 16 outcomes. (ii) Without replacement: S = {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)}, 12 outcomes. (iii) Sizes 16 and 12.

Check: 16 − 12 = 4, exactly the four ‘same number twice’ pairs ✓.

Answer to write in the exam

(i)

Tree: 1st draw 1, 2, 3, 4; from each, 2nd draw 1, 2, 3, 4 (as drawn)

∴ S = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4)}

(ii)

Tree: 1st draw 1, 2, 3, 4; from each, 2nd draw = the other 3 numbers (as drawn)

∴ S = {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)}

(iii)

(i) n(S) = 4 × 4 = 16

(ii) n(S) = 4 × 3 = 12

∴ The sample sizes are 16 and 12.

Common mistakes that cost marks

  • In (ii), still including (1, 1), (2, 2), (3, 3), (4, 4). Without replacement the same ball cannot be drawn twice.
  • Treating (1, 2) and (2, 1) as the same outcome. The order of the draws matters: first 1 then 2 is different from first 2 then 1.
  • Adding instead of multiplying: 4 + 4 = 8 or 4 + 3 = 7. Each first-draw branch has its own set of second-draw branches.

How this can come in the exam

MCQ (1 mark)

Cards numbered 1 to 5 are in a bag. Two cards are drawn one after the other without replacement, and the order is recorded. The number of outcomes in the sample space is

  1. 10
  2. 20
  3. 25
  4. 9
Show answer

(B) 20
5 choices for the first card, 4 for the second: 5 × 4 = 20.

Short answer (2 marks)

Three balls numbered 1, 2, 3 are in a box. One ball is drawn, returned, and a second ball is drawn. Write the sample space and find the probability that both numbers are the same.

Show answerS = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}, n(S) = 9. Same: (1, 1), (2, 2), (3, 3). P = 39 = 13.

Try one yourself

With the 4 numbered balls drawn without replacement (part ii), find the probability that the sum of the two numbers is 5.

Show answer

Favourable: (1, 4), (2, 3), (3, 2), (4, 1): 412 = 13.

More questions like this

All Probability questions · All maths questions

← Write the sample space and calculate the probability based on the giveList the elements of a sample space for the simultaneous tossing of a →