Transcript Probability Fancy a Flutter

Probability
Aim of session:
• Intoduce activities that explore and
develop ideas of probability
• Links with NZC and Progressions
• Links with National Standards and NZ
illustrations
Probability
How might we adapt this activity for our
Level 1 students? (refer to Progressions)
What is “everyday language”?
won’t
might
always no
Yes
perhaps
no way
maybe never
Impossible
certain
Will
Probability
How might we adapt this activity for our
Level 2 students?
Simpler / less ambiguous statements?
Measurable outcomes?
Some Content Knowledge!
• The Probability Scale:
– The probability of an event that is certain to happen is 1.
– The probability of an event that will never happen is 0.
– The probability for all other events is between 0 and 1. The
more likely it is to happen, the closer the probability is to 1.
Some Content Knowledge!
• Simple probability terminology:
- impossible, certain, likely, unlikely, even chance…
Some Content Knowledge!
• Probability of an event:
Number of favourable outcomes
Number of possible outcomes
But… count/list outcomes first.
Expressing probabilites as fractions not expected until Y8.
Some Content Knowledge!
• Variation: despite what is expected (from models
of of possible outcomes), actual results may differ
from this.
Some Content Knowledge!
•
Independence: what happens in one event, has no
affect on another.
Some Content Knowledge!
Sample size: the larger the sample, the closer the actual
results will be to the expected results.
Some Content Knowledge!
Be careful when interpreting data!
Juggling
Drinking and driving
Drinking
Driving
Good, rich tasks needed!
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Noticing distributions
Describing shape, centre and spread
Understanding variation
Acknowledging sample size
Fancy a Flutter?
A Horse Race game to develop
ideas of probability
Is it fair? Give your reasons.
Complete a chart showing how may times
each horse won a race.
Horse
Number
1
2
3
4
5
6
Which horse never moved? Why?
7
8
9
1
0
11 1
2
Complete a chart showing how may times
each horse won a race.
Horse
Number
1
2
3
4
5
Which horse won most often?
6
7
Why?
8
9
1
0
11 1
2
Use your data to investigate outcomes that
are possible and decide whether the game
is fair.
What are the possible outcomes?
Horse Number
1
2
3
4
5
6
7
8
9
10
11
12
Combinations
Total Number of
Combinations
What are the possible outcomes?
Horse Number
Combinations
1
2
(1,1)
3
(1,2) (2,1)
4
(1,3) (3,1) (2,2)
5
(1,4) (4,1) (2,3) (3,2)
6
(1,5) (5,1) (2,4) (4,2) (3,3)
7
(1,6) (6,1) (2,5) (5,2) (3,4) (4,3)
8
(2,6) (6,2) (3,5) (5,3) (4,4)
9
(3,6) (6,3) (4,5) (5,4)
10
(4,6) (6,4) (5,5)
11
(5,6) (6,5)
12
(6,6)
Total Number of
Combinations
0
1
2
3
4
5
6
5
4
3
2
1
Using a two-way table to find all possible
outcomes
Dice 1
1
Dice 2
1
2
3
4
5
6
2
3
4
5
6
Using a two-way table to find all possible
outcomes
Dice 2
Dice 1
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Models of All Possible Outcomes
• What is the total number of combinations
(outcomes)?
• Order the outcomes
• Probability of an event = Number of favourable outcomes
Number of possible outcomes
• Work out the probability for each horse to
move (express it as a fraction)
Expected Outcomes
After 36 Races
6
5
4
Wins
3
2
1
0
1
2
3
4
5
6
7
8
9 10 11 12
Experimental Results
• Play the game 36 times, 72 times…
(divide task between class)
• Graph results
• Compare to models of all possible
outcomes
The more we play the game, the
closer these two become? (Level 4)
Making the Task even Richer…
How could we make the Game Fair?
•Only use 6 horses
•Don’t include Horse 0
•Don’t include Horses that don’t move
very easily (0, 1, 12, 2, 11)
•Use a 12-sided dice
•Pull names out of a hat
•Adjust the race-course so that Horse 7
has to move 6 spaces to win,
Horses 6 and 8 have to move 5 spaces
etc.
What might we expect students
to do in order to meet each of
the standards?
In groups, develop criteria to help make
judgments in relation to the standards for
the Horse Race investigation.
Use the illustration “Dicey Differences”,
NZC Second Tier support, and your
copy of Mathematics Standards to help.
What might we expect students
to do in order to meet each of
the standards?
After 2 years: Identify all of the horses that
might win.
After 3 years: Identify which horses are
more likely and less likely to win.
By the End of Year 4
Identify that horse 7 has the best chance
of winning and that it’s impossible for
horse 1 to win.
Identify that horses 2 and 11 could still
possibly win, even though the other
horses are more likely.
By the End of Year 5
List the possibilities and order the
probabilities for horses to win correctly,
noting that e.g., horse 7 is “most likely”
to win, horses 6 and 8 have an “equal
likelihood”, horses 2 and 11 are equally
“most unlikely” to win and that it’s
“impossible” for horse 1 to win.
By the End of Year 6
Develop a model (e.g. 2-way table) to
show all possible outcomes. From the
model, explain that e.g. “there is only one
way for horse 2 to move”, etc.
By the End of Year 7
• Create a model of all possible outcomes and
identify, e.g. that horse 7 can move as a
result of 6 of the 36 possible outcomes.
• Predict that this outcome should occur about
once every 6 rolls of the dice, but recognise
that the actual experimental results are
unlikely to be identical to this.
• Recognise that their results may well differ
from their neighbour’s due to the variability
and independence of samples.
By the End of Year 8
• Organises results systematically.
• Creates a model (eg 2-way table) for all
possible outcomes.
• Expresses likelihoods as fractions,
concluding that e.g. the chance of Horse 7
moving is 1/6 but accepts that their results
may not exactly reflect this.
Cross the River
Place your 12 counters along the river bank.
You can place more than one counter on different numbers.
Roll two 1-6 dice and add the numbers. Move a counter across if you have on on that space
You are aiming to move your counters across the river before your opponent does.
9
1
2
8
3
4
5
6
7
10
11
12
Probability
Aim of session:
• Intoduce activities that explore and
develop ideas of probability
• Links with NZC and Progressions
• Links with National Standards and NZ
illustrations