Experimental probability - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Experimental probability

Probability

Lesson 3 of 6

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Probability

Lesson 3 of 6

Answer each one, then check.

1

Simplify 30/120.

2

Write 18/60 as a decimal.

3

What is the theoretical probability of heads on a fair coin?

4

Work out 500 × 0.3.

5

What is 1/6 as a decimal, to 2 decimal places?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Probability

Lesson 3 of 6

1

Simplify 30/120.

¼

2

Write 18/60 as a decimal.

0.3

3

What is the theoretical probability of heads on a fair coin?

½

4

Work out 500 × 0.3.

150

5

What is 1/6 as a decimal, to 2 decimal places?

0.17

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Probability

Lesson 3 of 6

1

Work out relative frequency.

2

Use relative frequency to estimate probability.

3

Predict how many times an event will happen.

4

Decide whether a coin, dice or spinner is fair.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Probability

Lesson 3 of 6

RELATIVE FREQUENCY

Relative frequency is the number of times an event happens divided by the total number of trials.

relative frequency = (frequency of the event)/(total number of trials)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Relative Frequency Settles Down

Probability

Lesson 3 of 6

The more trials, the closer the relative frequency gets to the true probability.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Relative Frequency

Probability

Lesson 3 of 6

A spinner is spun 60 times. It lands on red 18 times, blue 30 times and green 12 times. Find the relative frequency of each colour.

1

Red

18/60 = 0.3

2

Blue

30/60 = 0.5

3

Green

12/60 = 0.2

ANSWER

Red 0.3, blue 0.5, green 0.2. These add up to 1.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Predicting Outcomes

Probability

Lesson 3 of 6

Use the spinner results to estimate how many times you would expect blue in 500 spins.

1

Estimated probability of blue

0.5

2

Multiply by the number of spins

0.5 × 500 = 250

ANSWER

About 250 blues.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Is the Dice Fair?

Probability

Lesson 3 of 6

A dice is rolled 120 times and lands on 6 a total of 30 times. Does this suggest the dice is biased?

1

Relative frequency of a 6

30/120 = 0.25

2

Expected for a fair dice

1/6 = 0.167, so about 20 sixes

3

Compare

30 is much bigger than 20

ANSWER

The relative frequency 0.25 is well above 1/6, which suggests the dice may be biased, but more trials would give a more reliable conclusion.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Theoretical or Experimental?

Probability

Lesson 3 of 6

THEORETICAL PROBABILITY

• Worked out from equally likely outcomes.

• Example: P(head) = ½.

• Does not change.

EXPERIMENTAL PROBABILITY

• Estimated from the results of trials.

• Example: 18 heads in 40 tosses gives 0.45.

• Changes as you do more trials.

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How Reliable Is an Estimate?

Probability

Lesson 3 of 6

More trials give better estimates.

Small samples

10 tosses might give 7 heads: 0.7 is a poor estimate of 0.5.

Large samples

1000 tosses will usually give a relative frequency very close to 0.5.

Fair

A fair dice or coin should give relative frequencies close to the theoretical values.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Probability

Lesson 3 of 6

Trial

One repeat of an experiment.

Frequency

How many times an outcome happens.

Relative frequency

Frequency divided by the number of trials.

Theoretical probability

The probability worked out from equally likely outcomes.

Expected number

Probability multiplied by the number of trials.

Fair

Every outcome is equally likely.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Probability

Lesson 3 of 6

YOUR TASK

Coin Toss Experiment

15 minutes

Toss a coin 10 times and record the number of heads. Combine class results in groups of 10, 50 and 100 tosses. Work out the relative frequency of heads for each total and comment on how close each is to 0.5.

1

Collect results.

2

Combine.

3

Compare.

WHAT A GOOD ANSWER SHOWS

Small groups vary widely (for example 3 or 7 heads out of 10), while the combined class total is very close to 0.5. This shows that relative frequency settles as the number of trials grows.

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Can I...?

Probability

Lesson 3 of 6

Work out relative frequency.

Use it as an estimate of probability.

Work out an expected number.

Compare experimental and theoretical results.

Say whether a dice looks fair.

Explain why more trials are better.

Show relative frequencies add to 1.

Draw a relative frequency graph.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Relative frequency = frequency/trials.

Expected number = probability × trials.

More trials give a more reliable estimate.

Compare with the theoretical probability to test fairness.

EXAM FOCUS

A spinner is spun 200 times. It lands on red 62 times. Work out an estimate for the probability that it lands on red, and how many reds you expect in 500 spins. (3 marks)

Relative frequency is frequency divided by the number of trials. Multiply by the number of spins for the expected number.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Experimental Probability

Probability

Lesson 3 of 6

Answer all questions. Show your working. · 25 minutes

Question 1 · 2 marks · Calculator

A spinner is spun 60 times. It lands on red 18 times. Work out the relative frequency of red.

Question 2 · 3 marks · Calculator

A spinner is spun 200 times. It lands on red 62 times. Work out an estimate for the number of times it will land on red in 500 spins.

Question 3 · 3 marks · Calculator

The graph shows the relative frequency of heads as a coin is tossed more and more times. Use it to explain whether the coin is likely to be…

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Experimental Probability (cont.)

Probability

Lesson 3 of 6

Question 4 · 3 marks · Calculator

A dice is rolled 120 times. It lands on 6 a total of 30 times. Kim says the dice is biased. Is she right? You must show working.

Question 5 · 3 marks · Calculator

The probability that a bus is late is 0.15. Work out the expected number of late buses in 200 journeys.

Question 6 · 2 marks · Explain

Amy tosses a coin 10 times and gets 7 heads. Ben tosses the same coin 1000 times and gets 505 heads. Whose result gives a better estimate…

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 2 marks · Calculator

Probability

Lesson 3 of 6

“

A spinner is spun 60 times. It lands on red 18 times. Work out the relative frequency of red.

HOW TO ANSWER IT

Command word: Calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Probability

Lesson 3 of 6

2 marks available. Award a mark for each point made.

18/60

M1

0.3

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 3 marks · Calculator

Probability

Lesson 3 of 6

“

A spinner is spun 200 times. It lands on red 62 times. Work out an estimate for the number of times it will land on red in 500 spins.

HOW TO ANSWER IT

Command word: Calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

Probability

Lesson 3 of 6

3 marks available. Award a mark for each point made.

62/200

M1

0.31 × 500

M1

155

A1

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Question 3 · 3 marks · Calculator

Probability

Lesson 3 of 6

The graph shows the relative frequency of heads as a coin is tossed more and more times. Use it to explain whether the coin is likely to be fair. (3 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

Probability

Lesson 3 of 6

3 marks available. Award a mark for each point made.

The relative frequency settles

M1

It is near 0.5

M1

Conclusion that the coin is likely to be fair

C1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 3 marks · Calculator

Probability

Lesson 3 of 6

“

A dice is rolled 120 times. It lands on 6 a total of 30 times. Kim says the dice is biased. Is she right? You must show working.

HOW TO ANSWER IT

Command word: Calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Probability

Lesson 3 of 6

3 marks available. Award a mark for each point made.

30/120 = 0.25 or expected 20

M1

Comparison with 1/6 or 20

M1

A conclusion with a reason

C1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 3 marks · Calculator

Probability

Lesson 3 of 6

“

The probability that a bus is late is 0.15. Work out the expected number of late buses in 200 journeys.

HOW TO ANSWER IT

Command word: Calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

Probability

Lesson 3 of 6

3 marks available. Award a mark for each point made.

0.15 × 200

M1

30

A1

A sensible unit or statement

B1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 2 marks · Explain

Probability

Lesson 3 of 6

“

Amy tosses a coin 10 times and gets 7 heads. Ben tosses the same coin 1000 times and gets 505 heads. Whose result gives a better estimate of the probability of a head? Give a reason.

HOW TO ANSWER IT

Command word: Explain. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

Probability

Lesson 3 of 6

2 marks available. Award a mark for each point made.

Ben

B1

More trials give a more reliable estimate

C1

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