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IB Mathematics: Applications and Interpretation SL · 13.1: Probability · 13.1.10

Probability: exam review Exam review Worked solutions

Section A: short questions

  1. A game uses a fair spinner with 8 equal sectors numbered 1 to 8. Each spin is independent. [5]

    1. (a) Find the probability that a spin lands on a number greater than 5. [1]

      • A1 3/8 (0.375)

    2. (b) The spinner is spun three times. Find the probability that at least one spin lands on a number greater than 5. [2]

      • M1 for 1 − (5/8)^3
      • A1 387/512 (0.756)

    3. (c) The spinner is spun 200 times. Find the expected number of times that the spinner lands on a number greater than 5. [1]

      • A1 75 (200 × 3/8)

    4. (d) In a test, the spinner landed on a number greater than 5 on 36 of 80 spins. Write down the relative frequency of a number greater than 5 in the test. [1]

      • A1 9/20 (0.45) (36 out of 80)

  2. In a year group of 60 students, 34 study French and 27 study Spanish. Eight students study neither language. [5]

    1. (a) Find the number of students who study both languages. [2]

      • M1 for 34 + 27 − (60 − 8)
      • A1 9

    2. (b) A student is chosen at random. Find the probability that they study French only. [1]

      • A1 5/12 (0.417) (34 − 9 = 25 out of 60)

    3. (c) Find the probability that the student studies exactly one of the two languages. [2]

      • M1 for (25 + 18) ÷ 60, or 25 + 18 = 43 out of 60
      • A1 43/60 (0.717)

  3. A fair four-sided die, numbered 1 to 4, and a fair eight-sided die, numbered 1 to 8, are rolled together. The score is the product of the two numbers. The sample space diagram shows every score. [6]

    ×12345678112345678224681012141633691215182124448121620242832

    1. (a) Find the probability that the score is 12. [2]

      • M1 for counting the scores of 12 in the table (3 of 32)
      • A1 3/32 (0.0938)

    2. (b) Find the probability that the score is greater than 20. [2]

      • M1 for counting the scores above 20 (5 of 32)
      • A1 5/32 (0.156)

    3. (c) The dice are used 160 times. Find the expected number of times that the score is odd. [2]

      • M1 for P(odd) = 8/32 = 1/4, then 160 × 1/4
      • A1 40

  4. The two-way table shows how 118 students travel to school. Three of the numbers are missing. [7]

    WalksBusCyclesTotalBoy18271560Girl24??58Total4250?118

    1. (a) Find the three missing numbers. [2]

      • M1 for a method using the row and column totals
      • A1 23, 11 and 26

    2. (b) Find the probability that a student chosen at random is a girl who cycles. [1]

      • A1 11/118 (0.0932) (11 out of 118)

    3. (c) Given that a student chosen at random is a boy, find the probability that they cycle. [2]

      • M1 for restricting to the boy row (60 in total)
      • A1 1/4 (0.25) (15 out of 60)

    4. (d) Determine whether the events "boy" and "walks" are independent. [2]

      • M1 for P(boy) × P(walks) = 30/59 × 21/59 = 0.181 and P(both) = 9/59 = 0.153
      • R1 these are not equal, so the events are not independent

  5. A and B are events with P(A) = 0.55, P(B) = 0.4 and P(A ∪ B) = 0.82. [5]

    1. (a) Find \(P(A\cap B)\). [2]

      • M1 for 0.82 = 0.55 + 0.4 − P(A ∩ B)
      • A1 0.13

    2. (b) State whether A and B are mutually exclusive. Give a reason. [1]

      • R1 P(A ∩ B) = 0.13, which is not 0, so they are not mutually exclusive

    3. (c) C is an event that is mutually exclusive with A, and \(P(A\cup C) = 0.9\). Find \(P(C)\). [1]

      • A1 0.35 (0.9 − 0.55)

    4. (d) Find the probability that neither A nor B occurs. [1]

      • A1 0.18 (1 − 0.82)

  6. 80 people are asked about their transport. 18 own both a bike and a car. 11 own neither. 49 own a bike. [6]

    1. (a) Find the number of people who own a car but not a bike. [2]

      • M1 for 80 − 11 − 49
      • A1 20

    2. (b) A person who owns a car is chosen. Find the probability that they also own a bike. [2]

      • M1 for 18 out of 18 + 20 = 38
      • A1 9/19 (0.474)

    3. (c) A person who does not own a bike is chosen. Find the probability that they own a car. [2]

      • M1 for 20 out of 20 + 11 = 31
      • A1 20/31 (0.645)

  7. A family has three children. Assume that each child is equally likely to be a boy or a girl, independently of the others. [6]

    1. (a) Find the probability that exactly 2 of the 3 children are girls. [2]

      • M1 for listing the 8 equally likely outcomes, or using a diagram
      • A1 3/8 (0.375) (3 of 8)

    2. (b) Given that at least one is a girl, find the probability that exactly 2 are girls. [2]

      • M1 for removing the outcome with no girl (7 outcomes left)
      • A1 3/7 (0.429) (3 of 7)

    3. (c) Given that the oldest child is a girl, find the probability that exactly 2 are girls. [2]

      • M1 for keeping the 4 outcomes that start with a girl
      • A1 1/2 (0.5) (2 of 4)

  8. S and T are events with P(S) = 0.45, P(S ∩ T) = 0.18 and P(S ∪ T) = 0.67. [6]

    1. (a) Find \(P(T)\). [2]

      • M1 for 0.67 = 0.45 + P(T) − 0.18
      • A1 0.4

    2. (b) Find the probability that exactly one of the events S and T occurs. [2]

      • M1 for P(S ∪ T) − P(S ∩ T), or P(S) + P(T) − 2P(S ∩ T)
      • A1 0.49

    3. (c) Show that S and T are independent. [2]

      • M1 for 0.45 × 0.4 = 0.18
      • R1 this equals P(S ∩ T) = 0.18, so the events are independent

  9. A bag holds 6 green counters and 4 yellow counters. Two counters are taken at random, one after the other, without replacement. [5]

    1. (a) Find the probability that the two counters are the same colour. [3]

      • M1 for 3/5 × 5/9 (both green)
      • M1 for 2/5 × 1/3 (both yellow) and adding the two paths
      • A1 7/15 (0.467)

    2. (b) Given that the two counters are the same colour, find the probability that both are green. [2]

      • M1 for 1/3 ÷ 7/15
      • A1 5/7 (0.714)

  10. Three fair six-sided dice are rolled together and the three scores are added. [5]

    1. (a) Write down the number of equally likely outcomes. [1]

      • A1 216 (6 × 6 × 6)

    2. (b) Find the number of outcomes for which the sum of the three scores is 6. [2]

      • M1 for listing the combinations: 1+1+4 in 3 ways, 1+2+3 in 6 ways, 2+2+2 in 1 way
      • A1 10

    3. (c) Find the probability that the sum of the three scores is 6. [1]

      • A1 5/108 (0.0463) (10/216)

    4. (d) The three dice are rolled 540 times. Find the expected number of times that the sum is 6. [1]

      • A1 25 (540 × 5/108)

Section B: long questions

  1. A box holds 7 blue pens and 5 black pens. Two pens are taken at random, one after the other, without replacement. The tree diagram shows the situation. [12]

    7/12blue6/11blue?black5/12black?blue4/11black

    1. (a) Write down the two probabilities that are missing from the tree diagram. [2]

      • A1 5/11 (5 black among the 11 left)
      • A1 7/11 (7 blue among the 11 left)

    2. (b) Find the probability that both pens are blue. [2]

      • M1 for 7/12 × 6/11
      • A1 7/22 (0.318)

    3. (c) Find the probability that the two pens are different colours. [3]

      • M1 for 7/12 × 5/11 (first blue, second black) or 5/12 × 7/11
      • M1 for adding the two paths
      • A1 35/66 (0.53)

    4. (d) Given that the second pen is black, find the probability that the first pen is blue. [3]

      • M1 for P(second black) = 7/12 × 5/11 + 5/12 × 4/11 = 5/12
      • M1 for 35/132 ÷ 5/12
      • A1 7/11 (0.636)

    5. (e) A third pen is taken, also without replacement. Find the probability that at least one of the three pens is black. [2]

      • M1 for 1 − P(all three blue) = 1 − 7/12 × 6/11 × 1/2
      • A1 37/44 (0.841)

  2. Each of 70 students was asked whether they belong to the chess club (C), the drama club (D) and the music club (M). The Venn diagram shows the results, where x is a number. [13]

    14119x52x46CDM

    1. (a) Show that x = 7. [2]

      • M1 for 49 + 3x = 70
      • A1 x = 7 (AG)

    2. (b) Find the probability that a student belongs to exactly one of the three clubs. [2]

      • M1 for 14 + 11 + 9 = 34
      • A1 17/35 (0.486)

    3. (c) Given that a student belongs to Music (M), find the probability that the student also belongs to Drama (D). [2]

      • M1 for Music (M) in total = 32; Drama (D) and Music (M) = 14 + 4 = 18
      • A1 9/16 (0.562)

    4. (d) Find \(P(C\cup M)\). [2]

      • M1 for 70 − 6 − 11 = 53 (or P(C) + P(M) − P(C ∩ M))
      • A1 53/70 (0.757)

    5. (e) Two of the 70 students are chosen at random, without replacement. Find the probability that both belong to Drama (D). [3]

      • M1 for P(first in Drama (D)) = 36/70
      • M1 for P(second in Drama (D)) = 35/69
      • A1 6/23 (0.261)

    6. (f) Given that a student belongs to at least one club, find the probability that they belong to exactly two. [2]

      • M1 for denominator 70 − 6 = 64
      • A1 13/32 (0.406) (26 out of 64)

  3. At a college, M is the event that a student takes music and D is the event that a student takes drama. P(M) = 0.6, P(D | M) = 0.3 and P(M ∪ D) = 0.79. [12]

    1. (a) Find \(P(M\cap D)\). [2]

      • M1 for 0.6 × 0.3
      • A1 0.18

    2. (b) Find \(P(D)\). [2]

      • M1 for 0.79 = 0.6 + P(D) − 0.18
      • A1 0.37

    3. (c) Find \(P(M\mid D)\). [2]

      • M1 for 0.18 ÷ 0.37
      • A1 18/37 (0.486)

    4. (d) State whether M and D are mutually exclusive. Give a reason. [1]

      • R1 P(M ∩ D) = 0.18 ≠ 0, so they can both happen

    5. (e) Show that M and D are not independent. [2]

      • M1 for P(M) × P(D) = 0.6 × 0.37 = 0.222
      • R1 0.222 ≠ 0.18 = P(M ∩ D), so not independent

    6. (f) Three students are chosen at random. Assume that their choices are independent. Find the probability that at least one of them takes music or drama. [3]

      • M1 for P(neither) = 1 − 0.79 = 0.21
      • M1 for 1 − 0.21^3
      • A1 0.991

  4. Each day a student travels to school by bus, by bike or on foot. A fair spinner with three equal sectors decides how, so each way is equally likely. The probability that the student is late is 0.3 by bus, 0.1 by bike and 0.2 on foot. The tree diagram shows the situation. [13]

    1/3bus0.3late0.7on time1/3bike0.1late?on time?foot0.2late0.8on time

    1. (a) Write down the two probabilities that are missing from the tree diagram. [2]

      • A1 0.9 (the two outcomes of a branch add to 1)
      • A1 1/3 (three equally likely ways, 1/3 each)

    2. (b) Find the probability that the student travels by bus and is late. [1]

      • A1 0.1 (1/3 × 0.3)

    3. (c) Find the probability that the student is late. [3]

      • M1 for a product along a path, such as 1/3 × 0.3
      • M1 for adding the three paths: 1/3 × 0.3 + 1/3 × 0.1 + 1/3 × 0.2
      • A1 0.2

    4. (d) Given that the student is late, find the probability that the student travelled by bus. [3]

      • M1 for P(first way and late) = 1/3 × 0.3 = 0.1
      • M1 for dividing by P(late) = 0.2
      • A1 0.5

    5. (e) The student goes to school on three days. Assume that the days are independent. Find the probability that the student is late on exactly one of the three days. [2]

      • M1 for 3 × 0.2 × 0.8^2
      • A1 0.384

    6. (f) Find the probability that the student is late on at least one of the three days. [2]

      • M1 for 1 − 0.8^3
      • A1 0.488

jmaths.xyz · AI SL · 13.1.10 Unit review