Probability
Statistics 7: Probability
Must / Should / CouldKnow the words of probability, eg impossible, unlikely, even chance, likely, certain and relate them to a probability scale / Label a probability scale with fractions based on the number of parts it has been divided into / Express probability written as a fraction as a decimal or a percentage
Know the meanings of incident, event, trial, ‘favourable’ outcomes and ‘total possible’ outcomes
List all possible outcomes for a single event
Know how many cards there are in a pack / Write down the theoretical probability of an event based on the number of favourable outcomes compared to the total possible outcomes
Use and understand the terms, P(A) and P(A΄) / Understand, ‘mutually exclusive’
Know and use the fact that the sum of all mutually exclusive events is equal to 1; P(A) + P(A΄) = 1
Work out P(A΄) given P(A)
Understand the difference between theoretical and experimental probability
Record results for an experiment using a suitable table / Understand and work out ‘relative frequency’ as the total number of favourable outcomes compared to the total number of trials / Know how to use experiments to judge bias
Describe an experiment which will test for bias
Draw a relative frequency graph for an experiment
Understand that the most reliable relative frequency is obtained from the greatest number of trials / Use a relative frequency graph to work out the frequency of favourable outcomes after a specific number of trials
Use theoretic probability and relative frequency to predict the number of outcomes in populations which are simple multiples of the total outcomes / trials / Use theoretic probability and relative frequency to predict the number of outcomes in more complicated population sizes (requiring a proportionate calculation) / Extrapolate theoretical and experimental probabilities to prove or disprove a hypothesis
Understand the term, ‘combined events’
Use a space diagram or ‘two-way table’ to list all outcomes for combined events / Use a space diagram or 2-way table to work out the probability for combined events /
Know the difference between dependent and independent events / Draw a tree diagram for successive independent events
Calculate probabilities for independent events using the AND rule and OR rule / Draw a tree diagram for successive dependent events
Use a tree diagram to work out probabilities for all possible outcomes
Key Words:
Impossible, Very unlikely, unlikely, poor chance, evens, fifty-fifty, equally likely, likely, good chance, very likely, certain
Incident, event, trial, favourable outcome, total possible outcomes, mutually exclusive, bias, random, theoretical probability, experimental probability
Independent events, combined events, P(A), P(A΄), relative frequency, tree diagram, space diagram, two-way table
Plenaries:
Learning Framework Questions:
- What is the difference between an incident and an event?
- How do you turn an incident into a trial?
- What will the probability of an incident always be equal to?
- Can a probability of an event be greater than 1?
- What is the difference between theoretical probability and experimental probability?
- What does ‘biased’ mean?
- How can we work out ‘relative frequence’?
- What is the difference between dependent events and independent events?
Teaching Methods:
It is useful to read the MMMS script on probability (see MMMS scripts folder in Maths Shared Area) on the approach to introducing probability in Primary Schools.
Recommended sequence of teaching:
Understanding the words and notation
Theoretical Probability
Listing Possible Outcomes and Calculating Probability
Experimental Probability and Relative Frequency
Understanding Bias
Number of events v. probability graph
Calculating Expected Outcomes
Two Way Tables
One Event Tree Diagrams
Multiple Event Tree Diagrams
Using Tree Diagrams to Calculate Theoretical Outcomes
Understanding the words and notation of probability
Incident – something that happens
Outcome – a result of an incident
Trial – an incident where the outcome is recorded
Event – Something that has not happened yet (but you are thinking about it happening)
Students must have a clear idea of these definitions and how they further our understanding of probability. The following simple description can help to introduce the terms;
I am walking along a pavement when all of a sudden I trip up and fall over. This is an incident. A policeman is standing next to me. He doesn’t help! He just makes a note in his note pad that I tripped over. This incident has just become a trial. Now when I’m walking along a pavement, I keep thinking about tripping up and falling down in the future. This is an event, when you are ‘thinking about something happening which hasn’t happened yet’.
This story also allows for an introductory conversation about ‘outcomes’;
The possible outcomes of the incident of me tripping up and falling over could be;
- I hurt myself
- I get embarrassed
- I am no longer walking (or standing!)
- I have stopped walking … etc
This leads to a discussion of the question, ‘what is probability?’
By understanding the definitions of ‘incident’ and ‘event’, lead students to the understanding that the probability of an outcome if an incident has taken place (i.e. it has happened, we are not just thinking about it happening) is certain or impossible.
Consider the following example;
I flip a coin. It lands. What is the probability that the outcome is Heads?
The actual answer to this is ‘1’ or ‘0’ [i.e. certain or impossible], even though many will feel certain that it must be ½ or an equivalent value, since it has actually happened and is therefore an incident. The outcome is ‘predetermined’ and is already either Heads or Tails. Nonetheless, it is possible to ask a question to which the answer does give a probability of ½;
What is the probability that you will guess the outcome correctly?
Now, assuming the observer has not yet seen the outcome, the probability is indeed ½.
So, probability can be described as the likelihood of something happening which has not yet happened … it is about predicting the future.
Writing probabilities and using fractions:
Introduce the notation of P(A) where the whatever the desired outcome is, is written in brackets after P – for probability. Encourage students to read, ‘P(A)’ as ‘The probability of Event A happening’, therefore reinforcing the fact that probability is only relevant with events (as opposed to incidents). It is worth practising this notation by encouraging students to make sensible abbreviations for various defined events. For example;
The probability of the event of flipping a coin and getting Heads can be written as P(H)
The probability of the event of rolling a dice and getting greater than 2: P(>2)
The probability of the event of choosing prawn cocktail as a starter: P(PC)
and so on, and providing students with a range of scenarios for which abbreviations have been written in the form P(A) gives a further opportunity to consolidate use of this terminology.
Use the event of throwing a coin where the desired outcome is a Head
Introduce the
How many possible outcomes? 2
How many of these possible outcomes are favourable to getting Heads? 1
So, P(H) is 1 compared with 2
Written as: P(H) = [Resources: Probability words 10 Ticks Level 4 Pack 6 p27]
Probability scales and Theoretical Probability
When we introduce probability to students, we should make the distinction between ‘measuring probability’ and ‘estimating probability’. In order to ‘measure’ probability we have to be able to count equally likely outcomes and compare to the total possible number of outcomes, which reinforces the concept of ratio. This is not possible given certain contexts such as the weather which can only be ‘estimated’.
Students need to be able to draw a probability scale and describe probabilities in words and fractional values. They should understand that the length of the scale is unimportant as points along the scale (describing fractional parts such as 0, ¼, ½, ¾ and 1) are located proportionate distances along the probability scale.
Students should be able to draw scales and identify the following equivalences;
0 ⅛ ¼ ⅜ ½ ⅝ ¾ ⅞ 1
0% 25% 50% 75% 100%
0 0.25 0.5 0.75 1
Impossible / Very unlikely / Unlikely / Fairly unlikely / Even chance / Fairly likely / Likely / Very likely / Certain
In order to develop the idea of theoretical probability based on the number of favourable outcomes compared to the total possible number of outcomes, you can use the event of throwing a six-sided unbiased die:
Thinking about the event of rolling a die and getting the outcome: P(2)
How many possible outcomes? 6
How many of these possible outcomes are favourable to getting 2? 1
This is: ‘1 compared with 6’ which is written as: P(2)=
Ask students to consider varying the desired outcome, for example; P(even numbers), P(odd numbers), P(>2), P(<6), P(7) and compare the theoretical probability with words:
P(even number) = evens P(7)=0 impossible P(<7)=1 certain
Students must eventually understand the difference between theoretical probability and experimental probability. In order to do so, they need to understand that the theoretical probability of an event can only be calculated given a finite number of possible outcomes and given value of each outcome, assuming there is an equal weighting towards each outcome. It is therefore not possible to derive theoretical probability based on an event involving, for example, the weather or the number of goals scored in a football match.
Theoretical probability = number of favourable outcomes
Total number of possible outcomes
Resources: Living Worksheets: PROBAB03, PROBAB04
10 Ticks Level 5 Pack 1 p 39/40
10 Ticks Level 7/8 Pack 1 p3
Mutually exclusive events and the sum of probabilities
Ensure students understand that ‘mutual’ means 2 or more things have the same effect on each other, and ‘exclusive’ comes from the sense of exclude, meaning to prevent the other from doing something. So, mutually exclusive events are such that if the one outcome happens, the other outcomes cannot happen. This means that the sum of the probabilities for mutually exclusive events is equal to 1 (i.e.certain).
When throwing an unbiased six sided die:
Write P(1)= , P(2)= , P(3)= , P(4)= , P(5)= , P(6)=
Show +++++= 1
When throwing a coin:
Write P(H)= , P(T)= Show += 1
The sum of probabilities for all the outcomes of an event is 1
Then discuss things like if P(rain tomorrow) is what is P(no rain tomorrow)?
Resources: 10 Ticks Level 5 Pack 1 p 40 questions 10 to 14
10 Ticks Level 7/8 Pack 1 p 4
Listing Possible Outcomes and Expressing the Probabilities
Write all the possible outcomes of throwing two coins:
HH
HT
TH
HT
What is P(HH)? What is P(TT)?
Explain the difference between What is P(H or T)? When order is not
important
What is P(HT)? What is P(TH)? When order is
important
Tell the students to list all the possible outcomes of throwing three coins. Compare how many different possible outcomes there are. Discuss how students did this. Understand the importance of using a system. i.e. Changing only one thing at a time:
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
Ask the students to calculate the probabilities of various outcomes where order is important and order is not important.
For higher abilities repeat the exercise for four coins.
Resources: 10 Ticks Level 6 pack 6 p36/37
Experimental Probability:
Clearly there are other factors to take into account which favour a particular outcome and so the probability must be found by conducting experiments / trials.
Activity:
In pairs give the students a die. Tell the students to copy the table in their books
Score on Die / Tally / Frequency / Relative Frequency
1
2
3
4
5
6
Tell them to throw the die 60 times and every time it lands put a tally mark against the number it has landed on (this works really well if one pair is given a loaded die to work with, but try not to get them shouting out when they realise what is going on).
When they have completed 60 throws put their frequencies into a master table on the board (this works well when put into an Excel table on the projector)
Score on Die / Group Results / Total / Relative Frequency
1
2
3
4
5
6
Discussion from the Master Table:
Are any of the sets of results showing a bias towards a particular number?
What is bias?
How many times was an unbiased die thrown as a class?
How many times would you expect – a particular result for the 30 throws / for the whole class throws?
What is the theoretical probability for each outcome (fraction) and what is this as a decimal?
If I threw it ...... times how many times would I expect a 6?
Write the relative frequency in the last column and tell the students to write their relative frequency in their tables.
Experimental probability = number successful trials
(Relative Frequency) Total number of trials
Explaining Bias
What is bias?
What do you think if the referee at football match keeps giving penalties against your team when you haven’t done anything wrong?
Bias means ‘favours a particular outcome’
Unbiased means the result is random
Random means ‘all outcomes have a equal chance of happening’
Activity
Give the students the net of a die on card and a piece of blu-tac.
Cut out the net, stick the blu-tac on the inside of one of the sides, write their initials on a different side, stick it together. Then ask someone else in the room to throw it, record the outcomes to decide on which number it has been biased towards.
Record the results in a table like this:
Choose the biased number and plot the relative frequency against the number of trials.
Discuss the graph and what would it look like if more trials were done / if the results of another outcome were plotted, etc.
Resources: 10 Ticks Level 7/8 Pack 1 p11
Calculating Expected Outcomes
A starter of fractions of quantities questions would be a good lead in to this section.
Resources: 10 Tick Level 5 Pack 1 p41
Explain question 1 as an example
Two Way Tables:
Activity: Horse Racing.
Use a projector to project the race track on the board. 12 horses each have blu-tac on the back.
Give 12 students a horse each, they can choose by looking at the horses. Horse 1 at the top, 12 at the bottom in the start positions (all down the left hand side). Using 2 large dice, the rest of the class can throw them around the room; the sum of the 2 dice is the number of the horse that moves one space from left. The winner is the first one to the end of the race track. It doesn’t usually take long for students to observe horse 1 will not move.
At the end of the race observe the distribution of the horses and ask for why this may be.
Next, draw a sample space diagram showing all the outcomes of throwing 2 die.
Write the probabilities for all the possible outcomes and use this to demonstrate the distribution of the horses.
Resources: 10 Ticks Level 6 Pack 6 p 38/39
10 Ticks Level 7 Pack 1 p 13/14
Starters:
Generating words describing the outcome of an event and placing them on a probability line:
Objective of starter: To elicit prior knowledge
Description:
Higher: With a partner write words on scrap paper (3 or 4 minutes). Draw a line on the board & ask students to write their words on the line.
Equipment Required:None
Adding / Subtracting using a die to generate the numbers
Objective of starter: Includes use of a die & generate discussion regarding probabilities of outcomes, size of numbers & problem solving strategies.
Equipment Required:Die (preferably a large one to throw for all the class to see)
Description:
Draw boxes on the board like this:
And tell the students to draw it
(scrap paper or back of book)
Throw a die to generate 6 numbers to put in the boxes, write them on the board.
When the numbers have been put in ask the class to place the numbers in the boxes to make two 3 digit numbers
The tell them to do one of the following:
Add the two 3 digit numbers to make the smallest possible number.
Add the two 3 digit numbers to make the largest possible number.
Subtract the two 3 digit numbers to make the smallest possible number.
Subtract the two 3 digit numbers to make the largest possible number.
Once they understand what is going on tell the students what type of answer they are trying to achieve, then tell them to place the numbers in the boxes as they are generated. This will encourage them to consider the probabilities of the numbers being generated.
Any variation you like:
Discuss the strategies the students used for whichever variation you use.
Ladder numbers
Objective of starter: Encourages the students to think about the probabilities of outcomes of throwing a six sided die.
Equipment Required:Die (preferably a large one to throw for all the class to see)
Description:
Students draw 6 boxes in a column like this:
The aim is to place the score on the die, when thrown into one of the boxes and to place them all in numerical order, smallest to largest.
Variations:
More boxes
Use dice other than 6-sided.
Black & white counters game
Objective of starter: Encourages the students to think about the effect strategies for winning a game and collaboration with someone else.
Equipment Required:black and white counters (or any other colours)
Description:
Each student is given a black and a white counter.