Section 5 (4 Credits)
Random and Probability – It is all about
chance.
A random event has an unpredictable outcome.
There is a lot
of probabilistic language associated with probability and the
value we assign
to these words varies widely.
Task 1
(a)
What is the number that
represents an event that
MUST happen or be true. Eg You are alive yesterday.
(b) What
is
the number that represents an event that can never happen. Eg You
were dead
yesterday.
The actual numbers here are zero for “can never
happen” and
one for “must happen”. All other probabilities fall between 0 and
1.
A coin toss for example has two equally likely
outcomes.
Those are H or T. The probability of a head, Pr(H) = ½ = 0.5 = 50%
or “a half”
or “fiftyfifty”.
Language
Task 2
List 20 words in common use that have a
probability
associated with them. Eg Evens.
Here is a table to add your words and estimate
the
probability associated of them all.
Evens 
0.5 
Might 

Must 

Likely 

Never 

Pretty likely 

Probable 

Could 

Uncertain 

Possible 

Will 

Almost never 

Certain 

Almost Certain 

Half 

Unlikely 

Beyond reasonable chance 

On the balance of probabilities 













The curious thing about the estimates is
another person will
give you a different response. Is it a surprise to you that a
District Court
Judge estimated “beyond reasonable doubt” as 51%.
The language of probability includes words such
as “fair”,
“outcome”, “experiment”, “event”, “random generator”, “tool” and
“expectation”.
A spreadsheet is a wonderful place to explore probability using
the function
“RAND(0)”. This function generates a random number between 0 and
1.
The spreadsheet is actually a wonderful place
to explore
mathematics! It is quite surprising how many problems can be
solved using
probability on a spreadsheet as well. Simulating outcomes from
large runs on a
spreadsheet is used to solve some very difficult problems.
For example: If I toss three dice what is the
probability I
can form a triangle using these numbers as measures of the sides.
EG (1,1,6)
and (2,2,4) do not but (3,4,5) and (2,2,2) do.
A Statistical Investigation
All probability experiments should be shaped as
a
statistical investigation using the pseudonym PPDAC.
P = Problem
or Question
P = Plan or
method
D = Data or
the results of your plan
A = Analysis or
unpacking and
making sense of the data
C = Conclusion or
Answer to the Question
asked with elaboration.
A Simple Example of a Statistical Investigation
in
Probability
P = I wonder what the probability of throwing a
drawing pin
and having it land point up is.
P = I will toss the drawing pin onto a table
from about 30cm
and repeat this experiment 50 times. I am throwing it 50 times
because I think
that will give me a stable answer. The drawing pin could land on
its back or
side both of which are not point up.
D =
D,D,D,D,U,D,D,U,D,U,D,U,D,D,U,U,U,U,D,D,U,U,D,D,D
D,U,U,D,U,U,U,U,D,U,U,U,U,U,U,D,U,U,U,D,D,D,U,U,D
ANALYSIS
The number of D outcomes was 23 and the number
of U outcomes
was 27. The longest run of D’s was 4 and for U’s was 6. The total
number of
throws was 50. This makes the probability of throwing a drawing
pin point up
from this experiment 27/50 or 54%.
A graph of this data (using Excel) shows the
trending 54% of
“Pin Up” outcome. The graph starts to show stability and another
50 throws
would give a more reliable answer.
C = This investigation suggests the probability
of tossing a
drawing pin and having it land point up is about 54%. The graph
shows some
re\liability in this estimate as being slightly above 50%. The
margin of error
is 1/√50 = 14% suggesting a lot of variation is in the answer and
a better
estimate would be 54%±14% so the actual answer is very likely to
be between 40%
and 68%. This interval of error could be reduced by throwing the
pin many more
times. A good estimate would be about half.
Task 3
Create an experiment like this one and
investigate the
probability of an outcome.
Suggestions: “getting a 1 on a die, tossing
a head on a
coin, toast landing buttered side down”. Have some fun!
Returning to a task in Section 2
Imagine you are a coin and you flip yourself
100 times. Record
ten rows of ten outcomes you create. One the other grid toss a
coin 100 times
and record the outcomes. Give the coin a good toss [or use a die
and record
odds (H) and evens (T).]
This
task hopefully
illustrated how unrandom we are as humans. It would be a very
unusual person
who without prior knowledge guessed a long run of 5 or 6 Heads in
a row. The
overwhelming urge in all of us is to preserve a balance and
knowing that the
outcome is 50/50 we might get 2 or 3 in a row but very seldom more
than this.
Being random however allows for much longer runs. No notion of the
previous
outcome is taken into account for the next.
Games
A
die or two dice or more bring the element of chance to the table
and create an
astonishing number of very cool games and fun.
Monopoly
is the game to introduce buying and selling of property, tax,
penalties and
rewards. This is an essential game for all young minds.
Yahzee
is a number game using 5 dice. It involves getting patterns such
as runs of
5’s, a full house, two pairs and so on. Points are awarded for
scoring and a
total determines the winner.
This
score sheet can be enlarged and printed. A Yahzee is 5 numbers
of the same
type. Each player has three tosses and selects the dice to
retain before the
second and third throws.
Dungeons
and Dragons is a much more complex game using many different
dice and a complex
system of gaining and loosing strength. The goal being to
overcome the
opponent!
A
simple game using three dice to help develop number reasoning is
called
Skittles. The three dice are tossed and the resulting numbers
used to get the
answers 1,2,3,4,5,6,7,8,9 and 10. These are set up as skittles
are in a
triangle and crossed out when answers are created. EG 2,3,5 on a
dice gives
2x3–5 = 1.
7
8
9
10
4
5
6
2
3
1
Summary
Probability
extends into genetics, nuclear physics, chaos theory and
finance. Gaining a
working knowledge of a how a spreadsheet works will open up a
world of
solutions and fun.
Probability
language is a complex mix of creed and experience.
Lotto
is played by most Kiwis every week and they all know the chance
of winning is
less than 1 in 4 million. One in 100 is considered impossible. 1
in 4 million
is so screamingly small it is a wonder any one buys a ticket let
alone the
millions sold each week.
Horse
racing is plagued with cheats and casinos play knowing for all
the money that
is gambled about 4% must go to them. People continue to play the
odds.
Assessment
Section 5 (4 Credits)
Random and Probability – It is all about
chance.
xxxx
Find
BREAKOUT… a
Game for Two
Players
EQUIPMENT: Prison Block layout
below, ten counters or soft toy prisoners, 2x 1
to 6 dice.
COVID19 PRISON BLOCK
Cell
Block A 
Cell
Block B 
Cell
0 
Cell
0 
Cell
1 
Cell
1 
Cell
2 
Cell
2 
Cell
3 
Cell
3 
Cell
4 
Cell
4 
Cell
5 
Cell
5 
Cell
6 
Cell
6 
You can draw
this on a big
sheet of paper or set up the living room or even print this
sheet.
INSTRUCTIONS : Place your 10 prisoners
(counters/soft toys) in the
cells. You can spread them put or put them all in one or any
other combination.
One payer puts their prisoners in Cell Block A and the other in
Cell Block B.
THE FUN
PART
Now decide who
starts somehow
and taking turns toss both dice. Subtract the two
numbers to find the difference
(big – small) and if you have a prisoner(s) in that cell then let
one of
them out. For example if I throw a 3 and a 5, the
difference is 53=2 so I
release a prisoner form Cell 2 in my Block. Repeat until one
player has
released all his prisoners and wins the game.
HINTS: Play
the game a
couple of times to get the idea. Be a mathematician and look
for patterns.