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GCSE Maths · Higher · Chapter 13: Exploring and applying probability · Lesson 1

Experimental probability and expectation

Grades 4–7About 50 minutesNo calculator34 marks
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NameClassDate
A

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

  1. A1
    Needed today

    P(rain) . Find P(no rain).

    [1 mark]
    Answer
  2. A2
    From chapter 9

    A parallelogram has area 63 cm and base 9 cm. Find its height.

    [1 mark]
    Answer
  3. A3
    From chapter 3

    Pets owned: by 6 people, by 3, by 5. Find the total number of pets.

    [1 mark]
    Answer
  4. A4
    From chapter 8

    Make the subject of .

    [1 mark]
    Answer
B

Example, then your turn

Grades 4–5

Your teacher works through each example with you. Then try the one beside it.

Example 1

A spinner is spun 160 times. It lands on blue 52 times. Estimate the probability that it lands on blue.

Hint Relative frequency = number of times it happened ÷ number of trials.
  1. B1

    A paper cup lands on its side 78 times in 120 drops. Estimate, as a decimal, the probability that it lands on its side.

    [2 marks]
    Answer
Example 2

The probability that a seed grows is . 260 seeds are planted. Work out the expected number of seeds that grow.

Hint Expected number = probability × number of trials.
  1. B2

    A biased coin lands on heads with probability . It is thrown 225 times. Work out the expected number of heads.

    [2 marks]
    Answer
C

Practice

Grades 5–6
  1. C1

    Ali gets red 9 times in 30 spins. Bea gets red 25 times in 70 spins of the same spinner. Use all the results to estimate P(red).

    [2 marks]
    Answer
  2. C2

    A dice lands on 2 in 93 of 420 throws. How many 2s would a fair dice be expected to give? Is the dice likely to be biased?

    [2 marks]
    Answer biased?
  1. C3

    P(a battery fails in its first year) . 18 batteries failed, the expected number. How many batteries were sold?

    [2 marks]
    Answer
  2. C4

    A spinner lands on A, B or C. and . In 80 spins, how many Cs are expected?

    [2 marks]
    Answer
  3. C5

    A game at a fair costs £3 a go. The probability of winning is , and a winner is paid £8. 150 people play. Work out the expected profit for the stall.

    [3 marks]
    Answer
D

Exam-style questions

Grades 6–7
  1. D1

    Sara throws a coin 10 times and gets 7 heads. She says, "The coin must be biased, because a fair coin would give 5 heads."

    Explain why Sara is wrong.

    [2 marks]
  1. D2

    A bag holds 60 beads, each red, white or blue. Leah takes a bead at random, notes its colour and puts it back, 300 times. She gets red 55 times, white 125 times and blue 120 times.

    (a)

    Estimate the number of blue beads in the bag.

    [2 marks]
    Answer
    (b)

    Leah takes a bead 240 more times. Estimate how many times she gets a red bead.

    [2 marks]
    Answer
    Total for D2: 4 marks
  2. D3

    Kai throws a bottle top 100 times and records whether it lands open side up. After every 10 throws he works out the relative frequency of open side up so far. The graph shows his results.

    0204060801000.20.40.60.81.0Number of throwsRelative frequency of open side up
    (a)

    How many times did the bottle top land open side up in the first 20 throws?

    [1 mark]
    Answer
    (b)

    Use the graph to give the best estimate of the probability that the bottle top lands open side up. Give a reason for your choice.

    [2 marks]
    Answer reason
    (c)

    Kai throws the bottle top another 450 times. Estimate how many of these throws land open side up.

    [2 marks]
    Answer
    Total for D3: 5 marks
E

Extension

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

  1. E1

    A bag holds only red and green counters. A counter is taken at random and put back 400 times: red comes up 160 times. Then 10 green counters are added. In 300 more picks, red comes up 90 times. Estimate the number of red counters.

    [4 marks]
    Hint 1 · What to try

    Estimate the fraction of red counters before and after.

    Hint 2 · The first line

    With counters at first, red and also .

    Hint 3 · The full method

    , so and red .

    Answer
F

How did it go?

Colour one circle on each line.
Not yetNearlyYes
I can estimate a probability from an experiment or a relative frequency graph, and judge whether it is fair.
I can work out an expected number, and work back from one.

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