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

Probability: exam review Exam review: second paper Worked solutions

Section A: short questions

  1. A bag holds 12 identical discs numbered 1 to 12. A disc is taken at random, its number is noted and it is put back. Each draw is independent. [5]

    1. (a) Find the probability that a disc shows a number less than 5. [1]

      • A1 1/3 (0.333)

    2. (b) Four discs are drawn, one at a time, each put back. Find the probability that at least one disc shows a number less than 5. [2]

      • M1 for 1 − (2/3)^4
      • A1 65/81 (0.802)

    3. (c) A disc is drawn and put back 150 times. Find the expected number of times that the disc shows a number less than 5. [1]

      • A1 50 (150 × 1/3)

    4. (d) In a test, a number less than 5 came up on 21 of 45 draws. Write down the relative frequency of a number less than 5 in the test. [1]

      • A1 7/15 (0.467) (21 out of 45)

  2. A gym has 90 members. 52 members use the pool and 45 members use the sauna. Twelve members use neither. [5]

    1. (a) Find the number of members who use both the pool and the sauna. [2]

      • M1 for 52 + 45 − (90 − 12)
      • A1 19

    2. (b) A member is chosen at random. Find the probability that they use the pool only. [1]

      • A1 11/30 (0.367) (52 − 19 = 33 out of 90)

    3. (c) Find the probability that the member uses exactly one of the pool and the sauna. [2]

      • M1 for (33 + 26) ÷ 90, or 33 + 26 = 59 out of 90
      • A1 59/90 (0.656)

  3. A fair spinner with five equal sectors numbered 1 to 5 is spun and a fair six-sided die is rolled. The score is the product of the two numbers. The sample space diagram shows every score. [6]

    ×1234561123456224681012336912151844812162024551015202530

    1. (a) Find the probability that the score is 6. [2]

      • M1 for counting the scores of 6 in the table (3 of 30)
      • A1 1/10 (0.1)

    2. (b) Find the probability that the score is greater than 18. [2]

      • M1 for counting the scores above 18 (5 of 30)
      • A1 1/6 (0.167)

    3. (c) The spinner and die are used 150 times. Find the expected number of times that the score is odd. [2]

      • M1 for P(odd) = 9/30 = 3/10, then 150 × 3/10
      • A1 45

  4. The two-way table shows the favourite drink of 132 students. Three of the numbers are missing. [7]

    TeaCoffeeJuiceTotalYear 1131221770Year 1214??62Total4557?132

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

      • M1 for a method using the row and column totals
      • A1 35, 13 and 30

    2. (b) Find the probability that a student chosen at random is in Year 12 and prefers juice. [1]

      • A1 13/132 (0.0985) (13 out of 132)

    3. (c) Given that a student chosen at random is in Year 11, find the probability that they prefer juice. [2]

      • M1 for restricting to the Year 11 row (70 in total)
      • A1 17/70 (0.243) (17 out of 70)

    4. (d) Determine whether the events "Year 11" and "prefers tea" are independent. [2]

      • M1 for P(Year 11) × P(prefers tea) = 35/66 × 15/44 = 0.181 and P(both) = 31/132 = 0.235
      • R1 these are not equal, so the events are not independent

  5. E and F are events with P(E) = 0.62, P(F) = 0.35 and P(E ∪ F) = 0.85. [5]

    1. (a) Find \(P(E\cap F)\). [2]

      • M1 for 0.85 = 0.62 + 0.35 − P(E ∩ F)
      • A1 0.12

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

      • R1 P(E ∩ F) = 0.12, which is not 0, so they are not mutually exclusive

    3. (c) G is an event that is mutually exclusive with E, and \(P(E\cup G) = 0.95\). Find \(P(G)\). [1]

      • A1 0.33 (0.95 − 0.62)

    4. (d) Find the probability that neither E nor F occurs. [1]

      • A1 0.15 (1 − 0.85)

  6. 120 people are asked about pets. 25 own both a cat and a dog. 14 own neither. 68 own a dog. [6]

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

      • M1 for 120 − 14 − 68
      • A1 38

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

      • M1 for 25 out of 25 + 38 = 63
      • A1 25/63 (0.397)

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

      • M1 for 38 out of 38 + 14 = 52
      • A1 19/26 (0.731)

  7. A cow has four calves, born one at a time. Assume that each calf is equally likely to be a bull or a heifer, independently of the others. [6]

    1. (a) Find the probability that exactly 3 of the 4 calves are heifers. [2]

      • M1 for listing the 16 equally likely outcomes, or using a diagram
      • A1 1/4 (0.25) (4 of 16)

    2. (b) Given that at least one is a heifer, find the probability that exactly 3 are heifers. [2]

      • M1 for removing the outcome with no heifer (15 outcomes left)
      • A1 4/15 (0.267) (4 of 15)

    3. (c) Given that the first calf born is a heifer, find the probability that exactly 3 are heifers. [2]

      • M1 for keeping the 8 outcomes that start with a heifer
      • A1 3/8 (0.375) (3 of 8)

  8. U and V are events with P(U) = 0.6, P(U ∩ V) = 0.27 and P(U ∪ V) = 0.78. [6]

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

      • M1 for 0.78 = 0.6 + P(V) − 0.27
      • A1 0.45

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

      • M1 for P(U ∪ V) − P(U ∩ V), or P(U) + P(V) − 2P(U ∩ V)
      • A1 0.51

    3. (c) Show that U and V are independent. [2]

      • M1 for 0.6 × 0.45 = 0.27
      • R1 this equals P(U ∩ V) = 0.27, so the events are independent

  9. A bag holds 5 red counters and 3 blue 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 5/8 × 4/7 (both red)
      • M1 for 3/8 × 2/7 (both blue) and adding the two paths
      • A1 13/28 (0.464)

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

      • M1 for 5/14 ÷ 13/28
      • A1 10/13 (0.769)

  10. Three fair six-sided dice are thrown at the same time 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 7. [2]

      • M1 for listing the combinations: 1+1+5 in 3 ways, 1+2+4 in 6 ways, 1+3+3 in 3 ways, 2+2+3 in 3 ways
      • A1 15

    3. (c) Find the probability that the sum of the three scores is 7. [1]

      • A1 5/72 (0.0694) (15/216)

    4. (d) The three dice are rolled 432 times. Find the expected number of times that the sum is 7. [1]

      • A1 30 (432 × 5/72)

Section B: long questions

  1. A drawer holds 8 red socks and 6 grey socks. Two socks are taken at random, one after the other, without replacement. The tree diagram shows the situation. [12]

    8/14red7/13red?grey6/14grey?red5/13grey

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

      • A1 6/13 (6 grey among the 13 left)
      • A1 8/13 (8 red among the 13 left)

    2. (b) Find the probability that both socks are red. [2]

      • M1 for 4/7 × 7/13
      • A1 4/13 (0.308)

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

      • M1 for 4/7 × 6/13 (first red, second grey) or 3/7 × 8/13
      • M1 for adding the two paths
      • A1 48/91 (0.527)

    4. (d) Given that the second sock is grey, find the probability that the first sock is red. [3]

      • M1 for P(second grey) = 4/7 × 6/13 + 3/7 × 5/13 = 3/7
      • M1 for 24/91 ÷ 3/7
      • A1 8/13 (0.615)

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

      • M1 for 1 − P(all three red) = 1 − 4/7 × 7/13 × 1/2
      • A1 11/13 (0.846)

  2. Each of 80 students was asked whether they play football (F), hockey (H) and rugby (R). The Venn diagram shows the results, where x is a number. [13]

    168122x6x35FHR

    1. (a) Show that x = 10. [2]

      • M1 for 50 + 3x = 80
      • A1 x = 10 (AG)

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

      • M1 for 16 + 8 + 12 = 36
      • A1 9/20 (0.45)

    3. (c) Given that a student belongs to Rugby (R), find the probability that the student also belongs to Football (F). [2]

      • M1 for Rugby (R) in total = 31; Football (F) and Rugby (R) = 10 + 3 = 13
      • A1 13/31 (0.419)

    4. (d) Find \(P(F\cup R)\). [2]

      • M1 for 80 − 5 − 8 = 67 (or P(F) + P(R) − P(F ∩ R))
      • A1 67/80 (0.838)

    5. (e) Two of the 80 students are chosen at random, without replacement. Find the probability that both belong to Football (F). [3]

      • M1 for P(first in Football (F)) = 41/80
      • M1 for P(second in Football (F)) = 40/79
      • A1 41/158 (0.259)

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

      • M1 for denominator 80 − 5 = 75
      • A1 12/25 (0.48) (36 out of 75)

  3. In a city, R is the event that a commuter uses the rail line and S is the event that a commuter uses the shuttle bus. P(R) = 0.55, P(S | R) = 0.4 and P(R ∪ S) = 0.81. [12]

    1. (a) Find \(P(R\cap S)\). [2]

      • M1 for 0.55 × 0.4
      • A1 0.22

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

      • M1 for 0.81 = 0.55 + P(S) − 0.22
      • A1 0.48

    3. (c) Find \(P(R\mid S)\). [2]

      • M1 for 0.22 ÷ 0.48
      • A1 11/24 (0.458)

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

      • R1 P(R ∩ S) = 0.22 ≠ 0, so they can both happen

    5. (e) Show that R and S are not independent. [2]

      • M1 for P(R) × P(S) = 0.55 × 0.48 = 0.264
      • R1 0.264 ≠ 0.22 = P(R ∩ S), so not independent

    6. (f) Four students are chosen at random. Assume that their choices are independent. Find the probability that at least one of them uses rail or the shuttle. [3]

      • M1 for P(neither) = 1 − 0.81 = 0.19
      • M1 for 1 − 0.19^4
      • A1 0.999

  4. A shop sends each parcel by one of three couriers, chosen at random with equal chances. The probability that a parcel arrives damaged is 0.05 with courier X, 0.03 with courier Y and 0.04 with courier Z. The tree diagram shows the situation. [13]

    1/3X0.05damaged0.95not damaged1/3Y0.03damaged?not damaged?Z0.04damaged0.96not damaged

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

      • A1 0.97 (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 a parcel is sent by courier X and arrives damaged. [1]

      • A1 1/60 (0.0167) (1/3 × 0.05)

    3. (c) Find the probability that a parcel arrives damaged. [3]

      • M1 for a product along a path, such as 1/3 × 0.05
      • M1 for adding the three paths: 1/3 × 0.05 + 1/3 × 0.03 + 1/3 × 0.04
      • A1 0.04

    4. (d) Given that a parcel arrives damaged, find the probability that it was sent by courier X. [3]

      • M1 for P(first way and damaged) = 1/3 × 0.05 = 1/60 (0.0167)
      • M1 for dividing by P(damaged) = 0.04
      • A1 5/12 (0.417)

    5. (e) Four parcels are sent. Assume that the parcels are independent. Find the probability that exactly one of the four parcels arrives damaged. [2]

      • M1 for 4 × 0.04 × 0.96^3
      • A1 0.142 (0.141558)

    6. (f) Find the probability that at least one of the four parcels arrives damaged. [2]

      • M1 for 1 − 0.96^4
      • A1 0.151 (0.150653)

jmaths.xyz · AI SL · 13.1.10 Unit review