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Data and probability · Ages 14–15

Relative frequency and expected frequency worksheet

Part of: Probability

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Relative frequency and expected frequency worksheet

Data and probability · Ages 14–15

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Write each answer in the space. For word problems, show your working and write a sentence answer.

Part A: Working out and filling in (2 marks each)

  1. 1.Ka-yee rolls a die 500 times and gets a 6 on 58 of them. The relative frequency of a 6 = (as a decimal)

    Answer:

  2. 2.A spinner's pointer stops on red with probability 0.8. Spun 120 times, it should stop on red about times.

    Answer:

  3. 3.A spinner's pointer stops on red with probability 0.15. Spun 200 times, it should stop on red about times.

    Answer:

  4. 4.A bag holds red, yellow and blue balls. Tsz-hin draws one at random, notes its colour and puts it back, 500 times in all: red 77 times, yellow 107 times, and blue the rest. Blue's frequency is , and its relative frequency .

    Fill in each blank

    Answer:

  5. 5.A fair spinner has 8 equal parts, only 1 of them red. About how many spins should give red about 34 times?

    Answer:

  6. 6.A bag holds red, yellow and blue balls. Ming draws one at random, notes its colour and puts it back, 500 times in all: red 141 times, yellow 60 times, and blue the rest. Blue's frequency is , and its relative frequency .

    Fill in each blank

    Answer:

  7. 7.A fair spinner has 6 equal parts, 3 of them red. Spun 840 times, it stops on red 419 times. How far is the actual count from the expected one?

    Answer:

  8. 8.Tsz-ching tosses a coin many times and records the relative frequency of heads: 0.3 after 10 tosses, 0.43 after 100 and 0.49 after 1000. After 1000 tosses, heads had come up times; with the most reliable estimate, 200 more tosses should give about heads.

    Fill in each blank

    Answer:

  9. 9.Ka-lok tosses a coin many times and records the relative frequency of heads: 0.8 after 10 tosses, 0.57 after 100 and 0.482 after 1000. After 1000 tosses, heads had come up times; with the most reliable estimate, 1000 more tosses should give about heads.

    Fill in each blank

    Answer:

Part B: Choose, or true or false (2 marks each)

  1. 10.Four people each toss the same kind of drawing pin a different number of times and record the relative frequency of point up. Whose result gives the most reliable estimate of the probability of point up?

    • A. The one who tossed it 43 times
    • B. The one who tossed it 114 times
    • C. The one who tossed it 17 times
    • D. The one who tossed it 850 times

    Choose the right answer

    Answer:

  2. 11.A spinner has only red, yellow and blue. After many spins, the relative frequencies recorded are red 0.12, yellow 0.18 and blue 0.7. At least one of them must be wrong.

    True or false?

    Answer:

  3. 12.May rolls a die 600 times, and a 6 comes up 198 times. Which conclusion is the most reasonable?

    • A. Close to what was expected: no sign the die is biased
    • B. Too few rolls to conclude anything
    • C. The die must be fair
    • D. Far too many sixes: the die is probably biased

    Choose the right answer

    Answer:

Part C: Word problems (4 marks each)

  1. 13.In practice, Chi-keung's arrows hit the bullseye with a relative frequency settling at 0.9. In a match Chi-keung shoots 240 arrows. About how many should miss the bullseye?

    Working:

    Answer:

  2. 14.Ka-yee rolls a die 300 times: a 1 46 times, a 2 48 times, a 3 52 times, a 4 46 times, a 5 53 times, and a 6 the rest. A 6 came up times, a relative frequency of (as a fraction in its simplest form).

    Fill in each blank

    Working:

    Answer:

  3. 15.A and B each toss the same kind of bottle cap and count "open side up": A tosses it 12 times and gets 4; B tosses it 400 times and gets 46. A's result estimates the probability of open side up more reliably than B's.

    True or false?

    Working:

    Answer:

  4. 16.An estate has 8500 households. Of 50 visited at random, 13 keep a pet. Using the relative frequency, about how many households on the estate keep a pet?

    Working:

    Answer:

  5. 17.A stall's fair spinner has 8 equal parts: 3 red, 2 blue and the rest yellow. Spun 680 times, it should stop on red about times, and on yellow about times.

    Fill in each blank

    Working:

    Answer:

  6. 18.Ka-yee rolls a die 120 times: a 1 19 times, a 2 23 times, a 3 17 times, a 4 18 times, a 5 15 times, and a 6 the rest. A 6 came up times, a relative frequency of (as a fraction in its simplest form).

    Fill in each blank

    Working:

    Answer:

Show the answers and working

Relative frequency and expected frequency worksheet: Answers and working

Part A: Working out and filling in

  1. 1. 0.116

    Relative frequency = frequency ÷ number of trials = 58 ÷ 500 = 0.116.

  2. 2. 96

    Expected frequency = probability × number of trials = 0.8 × 120 = 96.

  3. 3. 30

    Expected frequency = probability × number of trials = 0.15 × 200 = 30.

  4. 4. 316, 0.632

    Blue: 500 − 77 − 107 = 316 times; relative frequency = 316 ÷ 500 = 0.632.

  5. 5. 272 spins

    P(red) = 18. Trials × 18 = 34, so trials = 34 × 8 = 272.

  6. 6. 299, 0.598

    Blue: 500 − 141 − 60 = 299 times; relative frequency = 299 ÷ 500 = 0.598.

  7. 7. 1 time

    Expected = 840 × 36 = 420; actually 419, a gap of 1.

  8. 8. 490, 98

    1000 × 0.49 = 490. The relative frequency after 1000 tosses is the most reliable: 0.49 × 200 = 98.

  9. 9. 482, 482

    1000 × 0.482 = 482. The relative frequency after 1000 tosses is the most reliable: 0.482 × 1000 = 482.

Part B: Choose, or true or false

  1. 10. D. The one who tossed it 850 times

    The more trials, the closer the relative frequency usually comes to the true probability: the one with 850 tosses has the most reliable estimate.

  2. 11. False

    All the outcomes' relative frequencies must add to 1: 0.12 + 0.18 + 0.7 = 1.

  3. 12. D. Far too many sixes: the die is probably biased

    A fair die rolled 600 times should give about 600 × 16 = 100 sixes; 198 came. Many rolls and a big gap make it probably biased; few rolls tell you nothing either way.

Part C: Word problems

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

  1. 13. 24 arrows

    The chance of a miss is about 1 − 0.9; expect 240 × (1 − 0.9) = 24. Or find the hits, 240 × 0.9, and take them from 240.

  2. 14. 55, 1160

    Sixes: 300 − (46 + 48 + 52 + 46 + 53) = 55; relative frequency = 55 ÷ 300 = 1160.

  3. 15. False

    The more trials, the closer the relative frequency usually comes to the probability: of 12 and 400 tosses, the larger number is more reliable, whichever relative frequency is bigger.

  4. 16. 2210 households

    The relative frequency of keeping a pet is 1350, taken as the probability: 8500 × 1350 = 2210.

  5. 17. 255, 255

    P(red) = 38: 680 × 38 = 255; yellow is 8 − 3 − 2 = 3 parts: 680 × 38 = 255.

  6. 18. 28, 730

    Sixes: 120 − (19 + 23 + 17 + 18 + 15) = 28; relative frequency = 28 ÷ 120 = 730.

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