1.1 sequences deﬁ nitely maybe - pearson education

34 Defi nitely maybe

1.1 Sequences

Unit objectives

• Interpret the results of an experiment using the language of probability; appreciate that random processes are unpredictable; interpret results involving uncertainty and prediction

• Understand and use the probability scale from 0 to 1; fi nd and justify probabilities based on equally likely outcomes in simple contexts

• Use diagrams and tables to record in a systematic way all possible mutually exclusive outcomes for single events and for two successive events

• Know that if the probability of an event occurring is p, then the probability of it not occurring is 1 – p

• Identify all the mutually exclusive outcomes of an experiment; know that the sum of probabilities of all mutually exclusive outcomes is 1 and use this when solving problems

• Estimate probabilities by collecting data from a simple experiment and recording in a frequency table

• Compare estimated experimental probabilities with theoretical probabilities, recognising that if an experiment is repeated the outcome may, and usually will, be different; and increasing the number of times an experiment is repeated generally leads to better estimates of probability

• Compare experimental and theoretical probabilities in a range of contexts; appreciate the difference between mathematical explanation and experimental evidence

Website links

• OpenerOffi cial government statistics

3 Defi nitely maybe

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Opener 35

Notes on the context

The transport and road traffi c accident data is taken from the Offi ce for National Statistics, the social trends report. For further information on the data, and to access other available statistics, please visit the relevant unit section at www.heinemann.co.uk/hotlinks.

The fi gures given for 2004 and 2005 are the most recent ones available at the time of writing.

Points to note, which could form the basis of a group discussion, include:

• In 2004, death rates for motorcycle users were over 40 times higher than for car users.

• From 1980 to 2004, the total number of road casualties fell by 14 per cent and the number of fatal and serious casualties fell by 60 per cent.

• In 2005, 52 per cent of road accident fatalities were for cars, 21 per cent pedestrians, 18 per cent motorbikes and 5 per cent cyclists.

• Pedestrian fatalities have fallen steadily since the mid 1990s. In 2005, there were 671 pedestrian fatalities, the lowest number for over 40 years.

• In 2005, the number of children killed or seriously injured in road accidents was 3500, a 49 per cent reduction on 1995 fi gures.

• Death rates for air travel were less than 0.1 per billion passenger kilometres from 1990 to 2004. Death rates for water travel were similar.

Discussion points

Discuss the trends in fatality rates over the 1981–2004 period. Identify factors that could account for a decrease in death rates – consider TV advertising campaigns such as those for drink driving.

What do the zeros mean in the data table? Is a risk of zero for water travel realistic?

Activity A

Motorcycles have the highest risk of fatal accident. Possible factors include the young average age of drivers (impulsivity and inexperience); less training needed to drive a moped or scooter, compared with driving a car; less protection in the event of an accident; less likely to be seen by other road users.

Activity B

Public transport (bus or coach, and rail), as a regular, form of transport, is the safest. Which form of transport is the safest overall? (According to the table, air is safest, but of course it isn’t an obvious option for most everyday travel.)

Answers to diagnostic questions

1

2

3 a) 3 _ 4 = 6 _ 8 = 12 __ 16 b) 5 _ 6 = 10

__ 18 = 45 __ 54

4 a) 3 _ 8 b) 2 _ 3

5 a) 63 b) 0.7

LiveText resources

• Leader of the pack

• Use it!

• Games

• Quizzes

• ‘Get your brain in Gear’

• Audio glossary

• Skills bank

• Extra questions for each lesson (with answers)

• Worked solutions for some questions

• Boosters

Level Up Maths Online Assessment

The Online Assessment service helps identify pupils’ competencies and weaknesses. It provides levelled feedback and teaching plans to match.

• Diagnostic auto-marked tests are provided to match this unit.

Impossible Unlikely Evens Likely Certain

b c aUNCORRECTED PROOF


3.1 What are your chances?

Objectives

• Use the vocabulary of probability when interpreting the results of an experiment; appreciate that random processes are unpredictable

• Use the vocabulary of probability in interpreting results involving uncertainty and prediction

Starter (1) Oral and mental objective

Display these fractions: 1 _ 4 , 2 _ 5 ,

7 __ 10 ,

3 _ 4 ,

3 _ 5 ,

1 _ 2 ,

3 __ 10 . Ask pupils to

put them in order, smallest to largest, and then convert the fractions to decimals (use prompts to help weaker pupils: How many tenths are there in a fi fth? What does three in the tenths column mean?) Check against the decimals to see if they have been ranked correctly.

Starter (2) Introducing the lesson topic

Display the following words: usual, defi nite, unlikely, impossible, even chance, probable, possible, good chance, uncertain.

Ask pupils to copy down the list in order – from least likely to most likely.

Discuss the order pupils have put them in – ‘impossible’, ‘defi nite’ and ‘even chance’ are probably the only words pupils will agree on.

Main lesson

– 1 Numbered probability scale

Remind pupils that the probability of an outcome of an event is the likelihood of the outcome happening and that probabilities can be written as numbers ranging from 0 to 1. Ask pairs of pupils to think of and discuss to think of and discuss different events. They should then list each of them under their appropriate heading (impossible, unlikely, even chance, likely, certain).

Discuss how the random nature of events makes them unpredictable, though it is possible to make conjectures about the likelihood of an event. How often does a coin land ‘tails up’? If it landed ‘heads up’ last throw, does it mean it will land ‘tails up’ next throw? Ask pupils to suggest outcomes for events that are defi nite, or impossible, and contrast these with random outcomes, for example: it is impossible to draw the fi fteen of hearts from a pack of cards, defi nite that you will draw either a black or red card but drawing the three of diamonds is a random outcome. Q1–2

– 2 Using simple fractions to describe probability

Review the use of simple fractions to give an estimate of the likelihood of an outcome in an experiment. Explain that fractions are used when we can use theory to work out the likelihood, for example: a dice has six sides so there is a one in six chance of it landing on any chosen number if the dice is fair.

– Compare fractions that have the same denominator. What is the likelihood of your birthday falling at the weekend/on a weekday? The larger numerator indicates the greater chance if denominators are the same. Q3

Resources

Main: a calendar

Activity A: a pack of cards for every eight pupils

Plenary: two dice

Intervention

Functional skills

Interpreting – pupils contrast descriptions of likelihood with precise assessment using fractions

Framework 2008 ref

1.1 Y8; 1.5 Y9; 5.4 Y8

PoS 2008 ref

Website links

www.heinemann.co.uk/hotlinks

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– Use a calendar to look at the likelihood of a day being a Monday in any one month. Compare months with the same number of days, then contrast months with different numbers of days to demonstrate that a chance of 4 __ 30 is greater than one of 4 __ 31 . This shows that if the numerators are equal, the smaller denominator indicates the more likely event. Use a simpler case to illustrate this – ask pupils to divide up a rectangles to demonstrate that 3 _ 4 is bigger than 3 _ 5 . Q4

– 3 Using fractions, decimaals and percentages tto describe probability

For simple denominators it may be possible to compare fractions with different numerators and denominators by fi nding the lowest common denominator, for example 5 _ 6 = 15

__ 18 and 7 _ 9 = 14 __ 18 . Use

a calculator to support the conversion of more diffi cult fractions to decimals when comparing complex probabilities. Q5–7

Activity A

This prediction activity emphasises that although it is possible to calculate the probability of an event occurring, the outcome is still random. If pupils have diffi culty remembering which cards have gone before, they could lay out the cards next to each other. Encourage them to work out the odds rather than guessing.

Activity B

Pupils order fractions. Differentiation: Increase the number of cards, or allow a wider range of numbers.

Plenary

Show two dice. If I add the score on the two dice, is it possible to make a number greater than three? Will I defi nitely make a number greater than three? What numbers is it impossible to make? What is certain about the result? Throw the dice and add the score. Am I likely to score higher/lower next throw? Why? Can I be sure this will happen? Why not?

Homework

Homework Book section 3.1.

Challenging homework: Devise a probability game using either a pack of cards or a pair of dice. Explain how to use probability to calculate the chances of winning the game.

Answers

1 a) i) 0.5 ii) 1 iii) 0b) Unlikely

2 a) Yes, there are 13 red cards left in the pack.b) Sabrina

3 a) A b) Any design with half the squares containing a sun

4 a) Quarter-fi nals b) Semi-fi nals

5 a) No b) Round 1

6 Bag A

7 a) Tirone b) No, because 29 ___ 833 = 0.03481… is greater than 19

___ 548 = 0.03467… .

What are your chances? 37

Related topics

Lifestyle choices can change the probabilities of dying from certain diseases. Insurance companies often require you to give information about smoking, drinking, eating and exercise habits before deciding the price of a life insurance policy.

Common diffi culties

Pupils may assume 24

__ 35 is greater than

7 __ 10 simply because both the numerator and denominator are bigger. Although using a calculator to convert them to decimals is easier, using the equivalent fractions 48

__ 70 and 49 __ 70 leads

to better understanding.

LiveText resources

Explanations

Booster

Extra questions

Worked solutions

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3.2 Representing probability

Objectives


• Find and record all the possible mutually exclusive outcomes for single events and two successive events in a systematic way using diagrams and tables


Display the following numbers: 300, 24, 120, 18, 3600. I have a one in three chance of winning at a game. How many times would I expect to win if I played 300 times, 24 times…? What if the chance of winning was increased to 50%? If I played 300 times and won 75 times, what are the odds of winning? (…as a fraction? …as a percentage?)


Display a probability line. Think of an event that you could place at this end, …the other end, …in the middle. Where would you place the chance of it raining tomorrow? Why can’t you agree?

Where would you place the chance of getting an even number when you roll a dice? Does everybody agree? Why?

Main lesson

– Make sure pupils understand that an ‘experiment’ is a situation involving chance or probability that leads to results called ‘outcomes’. An outcome is the result of a single trial for an experiment and an ‘event’ is one or more of these outcomes. For example, throwing a dice is an experiment, the outcomes are 1, 2, 3, 4, 5 or 6 and an event could be ‘throw a two’.

– 1 A probability scale

Explain that one way to visually indicate how likely an event is to occur is to place it on a probability line. Events that are impossible are assigned a value of 0, defi nite events are assigned a probability of 1. All other possible events are uncertain and the likelihood of their occurrence is indicated by a value between 0 and 1. Any question starting with ‘What’s the probability…?’ must have an answer that lies within the limits of the probability scale (0 and 1).

Q1

– 2 Calculating liklihood (1)

– 3 Calculating liklihood (2)

Explain that the likelihood of an event is the number of desirable outcomes divided by the number of possible outcomes. This information can be worked out using theory (a six-sided dice has six equal chances of landing on any one face – six outcomes) or by reference to information in charts, tables, lists or graphs. For pie charts pupils should appreciate it is necessary to

Resources

Activities: scissors; card; playing cards and a dice for each pair

Plenary: a coin and a six-sided dice

Intervention

Functional skills

Representing – identify number of outcomes using a sample space diagram Q5

Framework 2008 ref

1.1 Y8; 1.4 Y8; 5.4 Y8

PoS 2008 ref

Website links


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know the total represented by the full circle and the proportions for each event. With a bar chart the total should always be calculated fi rst by adding the heights of all bars. Probabilities are generally calculated as fractions, but decimals and percentages can also be used. Q2–3

– 4 Mutually exclusive outcomes

When calculating probabilities it is important for pupils to identify all the mutually exclusive outcomes fi rst. Outcomes are defi ned as mutually exclusive if they cannot happen at the same time. Q4

– 5 Sample space diagram

When an experiment is repeated, or followed by another one, a systematic search for all outcomes is advisable. This may take the form of a list, if only a few outcomes are possible, or a sample space diagram for a greater number of outcomes. The total number of outcomes possible can be confi rmed by multiplying the total number of outcomes for the fi rst experiment by the number of outcomes for the second (rolling a four-sided dice then a ten-sided dice would yield 4 × 10 = 40 outcomes and be represented by a 4 × 10 sample space diagram). Q5–6

Activity ACut two pairs of slots to feed the strips through, so that they can slide up and down. This activity can be used to illustrate the relatively small chance (1 in 9) of guessing correctly. It could lead to a discussion of the gambling odds. Consider the odds if the number of fruit options was raised.

Activity BA practical investigation of probability. Odds of winning: 5 __ 13 . Multiply the number of players by 13 to fi nd how many games you could expect to play before fi nding a winner.

PlenaryShow a coin and a six-sided dice. How many different outcomes are there if I roll them together? Explain that a ‘head’ is worth 1 and a ‘tail’ is worth 4 and that pupils need to add the coin score to the dice score to fi nd the total. Which is more likely, a total of four or fi ve? Why? What score is impossible? If I subtracted the two results, which total is most likely?

HomeworkHomework Book section 3.2.

Challenging homework: Investigate the odds for a fruit machine with three windows. Research the odds in other simple gambling activities.

Answers 1 a) D b) A c) B d) C

2 a) 1 _ 4 b) 1 _ 2 c) 1 _ 3 d) 1 _ 6

3 a) 1 _ 4 b) 1 _ 8 c) 1 _ 2 d) 1 _ 4 e) 0

4 a) Over 40/under 20, multiple of 8/multiple of 3, even/product of two odd numbers.b) Over 40, multiple of three, equal chance. c) 2 _ 3 ,

2 _ 3 , both 1 _ 2

5 a) A sample space diagram showing all nine outcomes. b) i) 1 _ 9 ii) 5 _ 9

6 Greater than 5 (fourteen of the outcomes are greater than 5 and only six are less than 5).

3.2 Representing probability 39

Related topics

Tie in with Citizenship themes and look at the risks involved with gambling. The odds are rarely understood. If you played the same set of six numbers for the lottery every time you would have to live to be 135 000 years old to have an even chance of winning.


Pupils tend to assume that repeating an experiment doubles the number of outcomes, especially if they have only considered events with two outcomes. Try not to over-practise simple examples; use a sample space diagram early on.

LiveText resources

Explanations

Booster

Extra questions

Worked solutions

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Objectives

• If probability of an event occurring is P then the probability of it not occurring is 1 − P

• Identify all the mutually exclusive outcomes of an experiment; know that the sum of mutually exclusive outcomes is 1 and use this when solving problems


Quick fi re fractions between 0 and 1. Pupils should reply with the complement to one. What do I need to add to four fi fths to get one whole?

Repeat with decimals (1 d.p. then 2 d.p.). Extend to fi nding complements of percentages up to 100%. Finally, get pupils to test each other, with a random selection of fractions, decimals and percentages.


Draw a probability scale. Pupils add a cross where they would estimate the chance of snow on Christmas Day (or a similar unlikely event). Ask pupils how many times it has snowed on Christmas in their lifetime and relate it back to where they have placed their cross. Explain that the proportion of the line from the beginning to the cross is directly related to the fraction they calculated.

Main lesson

– 1 Probability shorthand

Introduce the notation used for probability: use P(n) instead of the phrase ‘The probability of event n happening…’. Use the new notation to reinforce the use of probability scales.

Remind pupils that if an event has two mutually exclusive outcomes, then the probabilities of both outcomes must sum to one. Hence, deduce that if you know the probability of one outcome, the probability of the other outcome can be calculated by subtracting the fi rst probability from one. The probability of A not happening = 1 − P(A). Practise fi nding fraction, decimal and percentage complements to one. Use the probability scale to show the probabilities summing to one. Q1–2, 4

– 2 It adds up to 1

Extend the idea that the probabilities of mutually exclusive outcomes sum to 1 to work out the probability of an event if the sum of all other outcomes is known. If event A or event B can happen, then P(A or B) = P(A) + P(B). Q3

– Review sample space diagrams (see Lesson 3.2) as a way to identify all outcomes for two successive experiments with equally likely outcomes and hence calculate probabilities. Could you draw a sample space diagram to fi nd all the possible outcomes of fl ipping a coin three times?

3.3 It all adds up to 1

Resources

Activity A: pack of playing cards per pair

Activity B: prepared menu option cards (optional)

Intervention

Functional skills

Representing – probability notation simplifi es the situation by using a common set of symbols: P (n) Q3

Framework 2008 ref

1.1 Y9; 1.2 Y9; 5.4 Y8/9

PoS 2008 ref

Website links


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– 3 Tree diagrams

Introduce tree diagrams as an alternative approach. Use examples with equally likely outcomes as tree diagrams are used here merely to identify the number of combined outcomes. (Combined outcomes with unequal chances of happening are dealt with later.) Emphasise that, unlike sample space diagrams, tree diagrams can be used for more than two successive experiments but are cumbersome if there are too many outcomes at each branch. What would a tree diagram look like for a six-sided dice rolled three times? Explain that each branch of the tree diagram represents independent outcomes. Are you less likely to get a six the second time if you roll a six the fi rst time? Q5–6

Activity A

This activity gives pupils an opportunity to calculate and interpret probabilities but also demonstrates that if events are independent, knowledge about one probability cannot necessarily be used to deduce knowledge about the probability of a successive event – in contrast to mutually exclusive events, where the probability of an outcome can be deduced if all other probabilities are known.

Activity B

Pupils work with probabilities of combined events. To fi nd out whether their meals have a 25% chance of summing to £15 pupils should draw a tree diagram. As there are twelve possible outcomes only three of these should sum to £15. It could be made easier by preparing option cards, for example: starters at £3.00/£3.25/£3.50; main courses at £6.00/£7.00/£8.00; desserts at £4.00/£4.75/£5.50.

Plenary

Read out a ‘pessimist’s weather forecast’. There is a 10% chance of rain tonight with a 30% chance of frost early tomorrow morning and a 40% chance of further rainfall by the evening. Demonstrate how the ‘optimist’s weather forecast’ might sound. There is a 90% chance it will stay dry tonight with a 70% chance tomorrow morning will be frost free and a 60% chance it will remain dry until the evening. Repeat with different percentages, asking pupils to respond with optimist’s percentages.

Homework


Challenging homework: Make up your own glossary to explain the keywords used in probability. Use websites to make sure defi nitions are accurate.

Answers 1 a) 0.75 b) 0.25 c) 0.95 d) 0.75 e) 0.25 f) 0.05 g) 1

2 a) 1 _ 5 b) 3 _ 8 c) 3 _ 5 d) 4 _ 5

3 a) 0.2 b) 0.8 c) 0.4 d) 0.75

4 55%, 0%, 15%, 10%, 25%, 10%

5 a) Pupils’ tree diagrams b) No, there is a 1 _ 4 chance since the other possibilities are SN, NS or NN

6 a) 1 _ 8 b) 1 _ 8 c) 1 _ 4 d) 7 _ 8

3.3 It all adds up to 1 41


When summing to 1 there is a tendency to add the two columns independently for two-digit numbers and hence sum to 110% or 1.1, for example 35% and 75%, or 0.45 and 0.65 where both columns sum to ten. What do you get if you add the 3 and the 7 in the tens columns? How can it add up to 100% already if you haven’t added the units column yet?

LiveText resources

Explanations

Booster

Extra questions

Worked solutionsUNCORRECTED PROOF


3.4 Experimental probability

Objectives

• Collect data from a simple experiment and record in a frequency table; estimate probabilities based on this data

• Estimate probabilities from experimental data; understand that: if an experiment is repeated there may be, and usually will be, different outcomes; increasing the number of times an experiment is repeated generally leads to better estimates of probability


Display these numbers: 1 _ 3 , 25%, 1 _ 2 , 20%, 3 _ 4 .

An event occurs ten times out of forty. Which is the best estimate for the probability of it occurring again? Give more examples, using small numbers that cancel easily (4 out of 12). Extend to larger numbers that only approximate to the given numbers (13 out of 40).


Place nine cubes of one colour, and one of a different colour, in a box. Tell pupils the box contains 10 coloured cubes and they have to estimate the chances of taking out two cubes the same colour. Draw out two cubes and replace them several times. How many colours do you think are in the box? How many of each colour? Can you be certain? Would it help if I did the experiment another ten times?

Main lesson

– 1 Recording experimental data

Refer to other experiences of data collection. How many columns does a frequency table have? What are the titles? What is a better label for the total column? Describe the use of a tally and its advantages – check pupils fi nd the total frequency by adding in fi ves or tens. Remind them to total the frequency column and check it equals the number of times the experiment was carried out (sometimes referred to as the number of trials).

– 2 Estimating from experimental data

Estimates of probability can be made from data collected in experiments. However, pupils need to understand that a) the random nature of events means the frequency of a particular outcome may vary if an experiment is repeated and b) the reliability of the prediction can be increased by repeating the experiment more times. A simple coin or dice simulator will allow you to demonstrate this. Q1–3

– Estimating probabilities from combined events is much harder and pupils tend to be surprised by results of even simple experiments. What is the largest/smallest/most common result I can get if I add the numbers on two dice? Ask pupils to predict the probability of a combined outcome before conducting an experiment and then compare this with the experimental results. Rarely

Resources

Starter: coloured cubes, box

Main lesson: online coin or dice simulator

Activity A: paper, scissors, glue, counters

Activity B: dice per pupil

Plenary: coloured cubes, box

Intervention

Functional skills

Analysing – pupils should appreciate the value of collecting data to make projections about the future

Framework 2008 ref

1.1 Y8; 1.4 Y9; 5.4 Y8

PoS 2008 ref

Website links


UNCORRECTED PROOF

does experimental data throw up ‘easy’ fractions, so practise approximating (21 out of 80 is as good as 1 _ 4 ). If a calculator is used to divide the frequency by the total it is easy to see if the result is closer to common decimals. Q4

– 2 It adds up to 1

Explain that the probability estimates for combined events will enable you to make general predictions. They also let you predict the number of times an event is likely to occur. If there is a 25% chance of two spinners landing on red, how many times would you expect that to occur if you spun the spinners 50 times? Q5–6

Activity A

Pupils investigate the effect of bias. If you have biased dice, these can be used instead of the dice made from nets. To test the reliability of the estimate pupils can ask another pupil to do the same experiment with their dice and compare results.

Activity B

Pupils investigate the outcomes from a combined experiment. There are three prime outcomes (2, 3 and 5) and three that are square (0, 1 and 4). It appears to be a fair game but a sample space diagram reveals that P(prime) = 4 _ 9 and P(square) = 5 _ 9 . ‘Square’ has a slight advantage.

Plenary

Place three red cubes, two green cubes and one yellow cube in a box. Tell pupils there are six cubes in the box. Draw out a cube and replace it, six times. How many colours are there? Could there be a silver one in the box? Can you be sure? Repeat 30 times. Can you guess how many cubes are in the box and how many of each colour? If I did this another hundred times, how many of each colour would you expect? Reveal the contents of the box. How did you work out your probabilities?

Homework


Challenging homework: Modify Activity B to make it a fair game.

Answers 1 a) 2 b) 5 c) Yes, because a 2 came up 60

___ 200 times = 30%.

2 a) P(win) = 1 _ 5 , so you would predict two prizes but the event is random so either (or neither) result could occur.

b) The events are independent so the chances are still 1 _ 5 .

3 a) Tamsin carried out more trials. b) 12 __ 96 = 1 _ 8

4 a) 27 b) 100 c) 1 d) 27 ___ 100 , which is roughly 1 _ 4 .

5 a) 250 b) Answers around 250.

6 a) 1 _ 6 b) 60

3.4 Experimental probability 43

Related topics

Lifestyle choices can change the probabilities of dying from certain diseases. Insurance companies often require you to give information about smoking, drinking, eating and exercise habits before deciding the price of a life insurance policy.


Pupils tend to underestimate the total number of outcomes from combined events. A sample space diagram is the quickest way to illustrate the actual number of outcomes.

LiveText resources

Explanations

Booster

Extra questions

Worked solutions

UNCORRECTED PROOF


3.5 Can you trust experimental probability?


Two adults and two teenagers are standing at a bus stop. As the bus approaches they form a queue. Only two people get on. How many ways could the four people line up? What is the probability that both teenagers get on?


Use a simulator to demonstrate throwing a coin. How many ‘heads’ would you expect after six throws? 10 throws? 60 throws? Increase the number of trials to illustrate how the actual number of ‘heads’ tends towards the expected number.

Use a simulator to demonstrate rolling a six-sided dice. Would you expect to get just one six if you rolled the dice six times? How many might you get? How many times do you think you need to do this before you get a six roughly one sixth of the total number of trials?

Main lesson

– 1 Recording experimental data

Illustrate how a small number of trials may produce experimental probabilities that are unreliable when compared with theoretical probabilities. Use this fact to remind pupils that if probabilities are calculated from experimental evidence, as large a number of trials as possible should be carried out.

Q1–2

– When outcomes are not equally likely, theoretical probability cannot be used and so experimental probability must be used. How could you predict the outcomes for a biased dice? How many times would you have to roll it to calculate reliable experimental probabilities?

– Experimental probabilities can also be used to deduce information about an unknown situation; for example deducing the colours of a known number of counters placed in a bag. If I placed four coloured counters in a bag and took out, and replaced, a single counter 100 times, how many yellow counters would you guess were in the bag if I pulled out a yellow counter 53 times? Q3

– Experimental probabilities can be used to test whether an experiment is fair if it is possible to identify a sample space of equally likely outcomes and calculate theoretical probabilities to compare against the experimental results. Q4–5

Objectives

• Estimate probabilities from experimental data

• Compare experimental and theoretical probabilities in a range of contexts; appreciate the difference between mathematical explanation and experimental evidence

Resources

Starter: internet access

Activities: dice

Plenary: a pack of cards

Intervention

Functional skills

Interpreting and evaluating – contrast theoretical probability with experimental probability based on the evidence of results

Framework 2008 ref

1.4 Y9; 1.5 Y9; 5.4 Y9

PoS 2008 ref

Website links


UNCORRECTED PROOF

Activity A

Pupils use a sample space diagram to help predict the outcome of a combined event. When throwing two fair dice there are six outcomes with a total of 7, and all other totals are less likely. Predicting a total of seven is therefore a good strategy, but only if the game is played many times. As the winner is the fi rst to reach 10 points the outcome of the game is more likely to be determined by luck than strategy. A discussion about who won, and why, might help develop an understanding of the disparity between experimental results and outcomes predicted from theory.

Activity B

Pupils use theoretical probabilities found from a sample space diagram to make a game work to their advantage. A game is fair if the theoretical probability multiplied by the points awarded for a particular outcome gives a constant value for all nominated events. To gain an advantage one event should be awarded a disproportionate number of points.

Plenary

Take a pack of cards, deal one, but do not replace it. Ask pupils to predict the suit of the next card. Which suit is least likely? Repeat 20 times, sorting the cards into suits so pupils can see them. Is it easier to predict the next suit now? What is the theoretical probability of it being a Heart? Can you guarantee it will be a Heart next time? Is it ever possible to be certain about which suit will be drawn? What do you think your chances are of correctly predicting the correct card in the next 20 cards? Use the activity to illustrate how, although theory can be used to explain which card is most likely each time, randomness means experimental results rarely agree with theory. Are the theoretical probabilities more, or less likely to agree with the experimental results as the number of cards reduces?

Homework


Challenging homework: There is a link between the number of times you need to repeat an experiment to get reliable predictions and the number of possible outcomes. Use a simulator to fi nd this connection.

Answers 1 a) Pupil’s own frequency table. b) Pupil’s own probabilities.

c) P(A) = 1 _ 6 , P(B) = 1 _ 2 and P(C) = 1 _ 3 d) Repeat the experiment more times.

2 The six-sided dice. There are six possible outcomes for the six-sided dice but twelve outcomes for the twelve-sided dice so there will be more data for each outcome with the six-sided dice.

3 a) 1 × 1999, 2 × 2001 and 1 × 2002 b) 1 _ 4 c) 9 d) 2001

4 a) 22 ___ 200 = 11% b) 10% c) 20

d) Yes, the experimental probabilities are close to the theoretical probabilities.

5 a) 7 months start with J, A or M but only 5 start with F, O, N, D or S.b) Possible, but not probable. c) Any solution with equal chances of winning.

Can you trust experimental probability? 45


Pupils do not always appreciate the ‘acceptable’ margin of error when deciding whether an experiment is fair. They should look at the deviation from the expected frequency as a proportion, as well as considering the number of trials.

LiveText resources

Explanations

Booster

Extra questions

Worked solutions

UNCORRECTED PROOF


3.6 The best holiday ever – probably


In a very polite school everybody shakes hands with everybody else in the room. Six people are in a room. What is the probability that the tallest person shakes hands with the shortest person fi rst?

Pupils can make probability more complex than it is. They can be easily sidetracked calculating the total number of handshakes (15) when they only need to look at one person (the tallest) and the number of handshakes he/she must make (fi ve).


The names of four people, including Mr Smith, are put in a hat. One name is drawn and that person is allowed to draw a prize from a box containing fi ve prizes, one of which is a bottle of champagne. What is the probability that Mr Smith wins the bottle of champagne? Remind pupils how to identify outcomes (lists, two-way tables, tree diagrams).

Main lesson

– The fact that probability estimates are more reliable with an increasing number of trials needs emphasising as pupils often struggle to answer questions like ‘How could she improve the accuracy of her estimate?’ Q1

– ‘Mutually exclusive outcomes’ is another diffi cult concept. How would you describe ‘mutually exclusive’ in your own words? Q2

– Most of these problems focus on being able to calculate probabilities or use probabilities to calculate the frequency of an event. These calculations rely on being able to identify the total number of possible outcomes and knowing that the probabilities of mutually exclusive events must sum to one. What’s the fi rst thing you need to do before you can calculate the probability? (Find the total number of outcomes.) Q3–4, 7

– Review the use of the notation P(A) to describe probability. What sort of diagram could you use to help people who like pictures to help them

Objectives

• Solve problems and investigate in a range of contexts – problems involving probability


• Identify all the mutually exclusive outcomes of an experiment; know that the sum of mutually exclusive outcomes is 1 and use this when solving problems

• Estimate probabilities from experimental data; understand that: if an experiment is repeated there may be, and usually will be, different outcomes; increasing the number of times an experiment is repeated generally leads to better estimates of probability.

Resources

Activities: card.

Intervention

Functional skills

Communicating and refl ecting – This is an opportunity for pupils to refl ect on their learning and discuss the maths that sits behind problem-solving situations.

Framework 2008 ref

1.2 Y9; 1.4 Y8; 1.5 Y9; 5.4 Y8/9

PoS 2008 ref

Website links


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understand and learn? How do you write that in mathematical shorthand? Q5

– Theoretical probabilities can only be calculated if all outcomes are equally likely which, in practice, rarely happens. Hence a discussion about whether an event is ‘fair’ usually means ‘do all outcomes have an equally likely chance of happening?’ Q6

Activity A

The activity looks harder than it is. The favourite destination must be on four, fi ve or six cards (it appears most often, and all destinations appear at least once). So, the task is to identify a card that comes up 1 _ 2 ,

5 _ 8 or 3 _ 4 of the time. After playing, discuss

whether 20 trials were suffi cient to calculate the probability of the most often occurring card and the number of cards with this destination.

Activity B

Pupils should identify the seven outcomes (2–8) before playing. It is possible to devise a strategy for winning this game initially (call 5 as the total), but the probability of each total shifts as cards are removed. Discuss ways of solving this problem (a 16 × 16 two-way table with lines deleted as each pair of cards is removed.) The game shows that although the probabilities can be calculated the outcome is more likely to be decided by luck than strategy.

Plenary

Tell pupils that their names have been put in a hat for a prize holiday. Pretend to draw a name and say the name of somebody not in the class. Ask fi ve questions about the names in the hat. What are your chances of winning? They need to know that their name is in the hat and how many names there are in total (you added names from other classes as well). I will offer two prizes and put the fi rst prize winner back into the hat. What is the probability that the same person wins both holidays?

Homework


Challenging homework: Write three starter problems that could be used to begin a lesson on probability. Include your answers.

Answers 1 a) Highest number of sample fl ights.

b) Zoopc) Only a small number of sample fl ights.

2 a) 6 b) 1 _ 3 c) 1 _ 6

3 a) 85% chance of fl ying out on time.b) 93% chance of fl ying back on time.c) 72% chance of having enough snow to ski in February.

4 There may not be an equal number of boys and girls, or an equal number of pupils chosen for work experience from the two schools.

5 40, 28, 100, 12

Related topics

Probabilities are used to calculate risk in many professions. The work of actuaries is totally based on assessing risk. The training required, and the fi nancial implications of wrongly assessing risk, means it is a very well-paid career.


The word ‘estimate’ is often interpreted by pupils as ‘guess’. Repeat an experiment two or three times to show how the estimated probabilities are different but close enough together to enable predictions to be made using them.

LiveText resources

Explanations

Booster

Extra questions

Worked solutions

3.6 The best holiday ever – probably 47

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Transport events

Notes on plenary activities

Part 2: What assumptions must be made to answer this question?

Parts 6 and 7: The importance of reading questions fully should be highlighted here – in particular the signifi cance of the words ‘or’ and ‘not’.

Part 9: This question could form an interesting discussion point.

Solutions to the activities

1

2 We assume that Mikey walks to school if it does not rain and that he travels by car if it rains. So, P(walk to school) = 65%

3 Bus 3 __ 10 , Other 1 __ 10 Child is more likely to travel by bus as 3 __ 10 is greater than 1 __ 10 .

4 Using equivalent fractions: Walk 4 __ 10 , Car/van 2 __ 10 , Bus 3 __ 10 , Other 1 __ 10 ; so the most popular method of travel is walking.

5 a) 1 _ 2 b) 50%

6 9 __ 10

7 3 __ 20

8 a) ‘Other’ mode of transport could be taxi, bicycle, moped, motorbike, van, etc.

b) 1 __ 20

9 The information in the tables does not compare ‘like for like’. Table 1 represents only one type of journey, i.e. travel to school. Table 2 represents all types of journey as it represents all distances travelled.

Answers to practice SATs-style questions

1

(1 mark each)

2 a) boys girls

can swim 12 15

cannot swim 2 3

TOTAL 14 18 (1 mark)

b) P(boy) = 14/32 = 7/16 (1 mark)

c) P(girl cannot swim) = 3 __ 18 = 1 __ 16 (1 mark)

3 a) P(black) = 1 _ 5 (1 mark)

b) She has added 5 black balls (2 marks)

B A

A B

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Transport events 49

4 Sample space diagram:

5 6 7 8

2 7 8 9 10

4 9 10 11 12

6 11 12 13 14

8 13 14 15 16 (1 mark for table)

a) P(total is less than 10) = 4 __ 16 = 1 _ 4 (1 mark)

b) P(total is more than 10) = 10 __ 16 = 5 _ 8 (1 mark)

c) P(total is exactly 10) = 2 __ 16 = 1 __ 18 (1 mark)

Functional skills

The plenary activity practises the following functional skills defi ned in the QCA guidelines:

• Select the mathematical information to use

• Use appropriate mathematical procedures

• Find results and solutions

• Interpret results and solutions

• Draw conclusions in the light of the situation

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1.1 sequences deﬁ nitely maybe - pearson education

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