13 Probability

Getting started

This is a spinner.

Each colour is equally likely.

Find the probability of green. Red
a
White
Find the probability of blue or yellow.
An unbiased 6-sided dice is thrown.
Work out the probability of getting Bluc
Yellow

b Green
C an even number

d less than 5.

3 Tomorrow at 11:00 it will be sunny, cloudy or wet.

and
The probability it will be sunny is 25%
the probability it will be cloudy
is 40%.

Find the probability it will be

wet.

are dropped on the floor

4 A large number of drawing pins
down.
87 land point up and 135 land point
of landing point up.
Work out the experimental probability

SCISsorsS
scissors"? It is a very
Do you know the game 'rock, paper, beats paper
well.
old game and is known by
other names as
the
show either a fist (rock),
Wo people simultaneously
forwards (scissors)
or an open

rst two fr pointing

oers
paper
hand (paei). rock and rock
beats beals rok
paper beats
Ssors buls paper, cut paper, paper wraps
CISsors. S is because scissors

Tock and 10ck blunts scissors draw

choose the same thing it is a

layers
Doth rock
either wins) and they play again. beats seissors

275
 13 Probability

This may scem a trivial game but in 2005 the Maspro

electronics corporation used it to decide whether togv r to
t oauction its S20 million collection of paintings to Soe
Christic's auction houses
Alice,
Christic's won with paper., alier taking the advice of lor n
the T1-ycar-old daughters of onc of the directors of thc co tcnd
so they
strongest,
Their argument was that for
bepinners, rock scems

should start
with paper.
to start with that.
Playingagainst a beginner. you
his game illustrates two methods of finding probabilities
er - 1S

One method is to say that cach dilerent play - rock, scissors, pep
has a
each Ouc
If the three are cqually likely,
Cqua Iikely. outcomes

probability of
Fiora and Alice realised that, for less experienced players, the outcoil
than.
of starting with rock is m o i
are not
equally likely. The probability

13.1 Calculating probabilities

In this section you will . . . Key word
find the probability of complementary events complementary
event
use lists and diagrams to show equally likely outcomes

use lists and diagrams of outcomes to calculate probabilities.

This is a spinner.
The probability that it points to red is 0.2.
The p:obabiiiiy 1hat it pointsto blue is 0.15.
Red
We can write those probabilities as P(red) =0.2 and P(blue)=0.15
The sum of the probabilities for all six colours is 1.
This means the probability the spinner does not
P(not red)=I - 0.2=0.8
point to red, Bhue
The probability ihe spinner does not point to blue.
P(not blue)=1 -0.15=0.85
Getting blue and not getling blue are
complementary
One of them must happen and they cannot events
both happen.
IC A is event andA
an Is the
then P(A')=1-P(A) complementary event,

276
 13.1 Calculating probabilities

Morkedexample 13.1a

he
hahility that it will be sunny tomorrow is 40%.
probabili

will not rain tomorrow 95%.

bability it Is

The that tomorrow

md
probability
the

be sunny
awillnot
i twillrain.

Answer

Pnotsunny)=I=P(sunny)=100%-40%=60%
Prain)=l-P(not rain)=100%-95%=5%

Worked eaemple 13.1b

6-sided dice are thrown.

Tuo unbiased

Find the probability of getting

both dice
athesame number
on

ba total of 6

a total of 9 or more.
Answer
a The diagram shows all X x x
possible outcomes.
5 XX

There are 36 outcomes

altogether.
The loop shows the
Outcomes with the same
number: (1, 1), (2, 2)and X

on.

There are 6 of them. T23 4 56

First dice
Theprobability iswhich is
equivalent to

277
 13 Probability

Continued
b This table shows the total for
cach outcome or example, 5
on
Five outcomes give a total of
the first dice and
6(shown by a blue loop) 7 3 on the sccond
The pobability is 2 4 S 7 gives a total of g
2 3 4 5
36
c
Using the same table as for F'irst dic
part b. tecn outcomes give
a total of 9, 10. 11 or 12
(shown by the red loop).

The probabilityis= 18

Exercise 13.1
T he that a football team will win a match is 0.3.
probability Tip
The probability that the team will draw is 0.1.
All dice in this
Work out the probability that the team will
exercise are
not win b not drawW
unbiased, 6-sided
lose d not lose. dice.
2 Tomorrow must be hotter, colder or the same temperature as today.
The probability it will be hotter is 55%.
The probability it will be colder is 25%
Work out the probability that it will
not be hotter
not be colder
not be the same temperature.
spinner has five colours on it.
The
prubability it shows green is 0.32.
obability it shows purple is 0.17.
T

itd theprobability
that the colour is
nol precn
b not purple.

278
 13.1 Calculating probabilities

T h e r ea r e
ee lots of coloured toys in a box. Here are the percentages
colours
of the
some
of

Colour
yellow orange red green

P e r c e n t a g e
15% 25% 30% 10%

add up to less than 100%?

Why do the percentages
a
takes a toy at random.
A child

b
Find the probability that the toy is
not orange ii not green

ii not red iv not yellow

the probability that
Tuo dice
are thrown. Find
Tip
a
both dice show 5 Use the diagram
shows a 5 and the other does not trom part a of
b one dice
neither dice shows a 5.
Worked Example
are thrown. The numbers are added together. 13.1b
Two dice
Draw a table to show all the possible outcomes.
a
total is
b Find the probability that the iv 9
ii 7 ii 12
i 3
this table of probabilities.
c Copy and complete
11 12
Total 2 3 4 567 8 9 10
Probability
are added together.
thrown. The numbers
Two dice are
that the total is
a Find the probability than 5
ii more

5 or less
i iv less than 10
iii 10 or more
a p r i m e number.

with a probability of
Find an event to check it is
b to a partner
correct.

answer to part b shows the possible

C Give your This table
thrown.
fair dice are
A fair coin and
a

outcomes. T3 stands for a

Dice taii orn the coin
2
4 5 and 3 on the dice.

Coin H H1
T
T3
the table.
and complete tthey all equally likely?
a Copy
o u t c o m e s
are there?Are
b Howm a n y

279
13 Probability

Find the
c
probability of
i 6 and a
tai ii 4 and a head
number less than 3.
ii a head and an even tail anda
number iv a

d Find the
probability of each of the events in part e no
e
Describe an event with a probability of
Give your answer to
part to neck.
e to a partner ce
9 Here are two
spinners.
a
The two spinners are spun. Draw a diagram to
show all the outcomes.
b Work out the probability that
both spinners show a l
i neither spinner shows al
ii both spinners show the same number Tip
V t h e spinners donot show the same numbel. Use a table like
The two scores are added together. the one in part
Draw a table to show the possible totals. b of Worked
d Find the probability that the total is example 13.1b.
i 4 5

ii not 7 iv a multiple of 3

v a factor of 12.
Now the spinners are multuplied.
scores on the Tip
Draw a table to show the possible products. The product
Find the probability of each of the diflerent possible products. is the result of
multiplying two
9 Find the probability that the product is
ii less than 6 numbers.
i 6 or more

iii an odd number iV an even number.

10 a Two fair coins are flipped.

Copy and complete this table to show the outcomes.

Second coin
H T
H HT
First coin
T

280
 13.1 Calculating probabilities

Arun says:
Read whal When you throw
b
nplain why Arun is not correct.
two coins there are three
Find the probability of outcomes. They are 2 heads,
2heads ii 2 tails 2 tails or a head and a tail.
tail. So the probability of 2 heads
a head and a
ii
is
to shoW the outcomes
Another way
hen two fair coins are thrown is a tree diagram.
fill in the
Copy the tree diagram and missing outcomes. First Second Outcome

e
Explain how the table in part a and the tree diagram in part d coin Coin
same outcomes.
show the
Three fair coins thrown. One possible outcome is HHH, a
are H *

head on all three coins. List all the possible outcomes in this way.
three fair
Draw a tree diagram to show the results of throwing HT
9
coins. Use it to check your answer to part f.
of ,H
When three fair coins are thrown, find the probability
.

h
i 3 heads ii 3 tails
ii not getting 3 heads iv 2 heads and I tail

1 head and 2 tails.

mathematician
Think like a
thrown.
outcomes when 4 tair coins are

11 Investigate the possible of different events

outcomes and find probabilities
find all
You should
the possible
trom Question 10 to help you.
Use your experience

with numbers on them.

12 Zara has three cards

order to make a 3-digit number.

side in a random
side by
She puts the cards have found them all.
numbers. Make sure you
a List all the possible
formed is
that the number
b Find the probability an even number
more than 400
an odd number
has four cards.
an extra
card. Now she
Lara adds

them sde by side to make a 2-digit mber.

numl
and places
at random
Zara takes two cards sne can
makE.
Make you sure you
have found them
m all.
numbers
c List all the possible number
that the 2-digit
d Find the probability is not 48 ii is an odd number
i is 48 includes the digit 2.
number
even
IV IS an

281
13 Probability

Now Zara takes

three cards at random and places them
ide
by side to make a
3-digit number
eList all the possilble numbers she can make
Findthe probablity that the 3-digit number s
an odd number ii less than 500.
ii n cven number

In this exercise have used different methods to 1in

you
Outoomes. What are they? Which do you prefer and Wny

Summary checklist
can find the probability of a complementaryevent outcomes.
can use a chart, a table ora list to find all possible
can use lists and diagrams of outcomes to calculate probabilities.

13.2 Experimental and

theoretical probabilities
In this section you will .. Key words

calculate experimental probabilities and compare them to experimental

theoretical probabilities. probability
theoretical
Youcan use cqually likely calculate
probabilities.
outcomes to probaility
is not possible you can do an experiment.
When this
A spreadsheet s uscd to simulate throwing a dice 200 times.
Here are the resulis of the experiment.

Score 1 23 4 5
Frequency3036 37 33 35 29
Erom the information in the table, we can work out the
experimental probabilities:
The experimental probability of 1 is=0.15
200
The experimental probability of 2 is =0.18
200
The expermental probability of an even number is 36 +33 +2998
200 2000.49

282
 13.2 Experimental and theoretical probabilities

hat cac
i n o w That cach umber is equally likely with a lair dice se we can alo
heoretical probabilitic
a e the
theoretical probabihty of I is0167 to 3d

theoretical probab of 2 is0.167 to dp

1he
theoretcal probability of an even umber is 0.5s
The

he
menial probabilitics and the theoretical probabilitics are
eyermmen
very
This shows that the sprcadsheet simulation is reliable.
m i a r

Worked example13.2)
what Marcus says
Rod

Event
2 heads 2 tails 1 head and 1 tail
Frequency 17 14 19
Ihavethrown
a Calculate the experimental probability of 2 coins 50 times.
cach outcome.
The results are in

b Calculate
the theoretical probability of this table.
each outcome.

Marcus has made up his results.

Marcus's teacher thinks answer. for your
What do you think? Give a reason

Answer
is
aThe experimental probability of 2 heads 50
4

The experimental probability of 2 tails is0.28

1 head and I tail is =0.38
The experimental probability of TT
outcomes: HH, HT, TH,
bThere arefour equallylikely
heads is=0.25
probability of 2
The theoretical
=0.25
probability of 2 tails is also
Thetheoretical TH
and I tail: HT or

ways to get head

I
h e a r e two

==0.5
probability is
heoretical probabilities are
not similar
and theoretical results.
CThe experimental have made up
his
if Marcus may
It looks as

283
13 Probability

Exercise 13.2
A learner throws ul s.
a coin 50 times. This lable shows the
i
LTHT|T T|HHTH
H T
TH H T H HH T
H H T H HHTH T H
TT TTHTI
TH HT TH TH T
a Use the first row of the table to calculate the
experimenta
probability of a head based on the first 10 throws.
b Use the first two rows of the table to calculate the
experimental probability of a head based on 20 throws.
a head
C
In the sanme way, find the experimental probability of
based on
i 30 throws ii 40 throws ii 50 throws.
d Compare the experimental probabilities you have found so
far with the theoretical probability of a head.
The learner throws the coin another 50 times. Here are the results.

H H H H T T T LH
T T T T T HH
T T H H H T T
H T H H HH T H H
HTTT|HH|HHT
Use the two sets of results to find the
of a head based on 100 throws. How close
experimental probability
is it to the
theoretical probability?
This spinner has 3 sectors.
The probability of red, P(red)= 0.6
The probability of white, P(white) =0.3 Bh
The probability of blue, P(blue) =0.1
Here are the results of 50
spins.
R WRR B White
WR W R W BB
R
B RR
RW R W R
R|R R W
R W
RR R RR RR R
R R
RRRR WRR
B
W
B
R
R
R

284
 13.2 Experimental and theoretical probabilities

ise Cach row to find an experimental probability of red based

on 10 spins

wo diflerent sels of
25 spins and use them to find the
ind
ermental probability of s
red,
Uscall S0 spins to find experimcntal probabilities of
white and bluc.
Here are the results of 800 spins
d

Colour red white blue

Frequency 489 218 93
se these results t find experimental probabilities for
each colour.

Read what Marcus says:

It is better to use a large

number of spins to work out

experimental probabilities.

answer.
for your
agree? Give a
reason
Do you
throwing six dice together
This question is about
at least one 6.
and seeing if there is a

Four learners each

threw six dice together
Here are their results
number of times.
Marcus Zara
Arun Sofia
20 40 50
Name 10
Number of throwS 9 36

Frequency of at
least one 6
lor
at least one 6
probability of
experimental
a Work out the
each learner. another experimental
resulls to get
sets of
Combine the four
b
probability. There was at least one SIX
simulated
500 throw
ows.

computer

333 times. probability

from this dat.
experimenlal
an
Work out

285
13 Probability

d n favt, the one d s

least
pobability of throwingat
theoretical
theoretical
\permental probabilitics with the
ubabiht
Activity 13.2
will needa dice.
Work with another learner
on this question. Each pa"
Design and carny out an experiment to answer this question

unbiased?
Is your dice

Betore you start, you need to decide:

how many times to throw a dice
how to record your data
probabilities
probabilities and theoretical
L m p a r e experimental based on your data.
Whte your plan before vou start. Give reasons for your conclusion

4 Work with one or more other learners on this question.

ratio ol the
ou learnt about the number n (pi) in Unit 8. It is the
cireumference ot a circle to its diameter.
The value of is a decimal that does
n not terminate and has no

pattern to its digits.

Here are the first 200 decimal places of n.
3.141 592 653 589 793 238 462 643 383 279 502 884 197
169 399 375 105 820 974 944 592 307 816 406 286 208 998
628 034 825 342 117 067 982 148 086 513 282 306 647 093
S44 609 550 582 231 725 359 408 128 481 117 450 284 102
01 938 521 105 559 644 622 948 954 930 381 96
Look at this statement:

All
the digits from 0to 9 are equally likely.

Devise and carry out an experiment to test this statement.

experimental probabilities and
e

theoretical probabilities.
compare them with
b Describe your experiment and your result.
Give a reason for your conclusion.
Look at the results of another
pair.
How do they compare with yours?

286
 13.2 Experimental and theoretical probabilities

You n e c d . da spreadsheet this question. You also need to

to answer

how to
it to
use generate random numbers.
w
Carry out a simulation to model throwing a coin 50 times.

Find the experimental probability of throwing a head and

compare it with the theoretical probability
Repeat part a another S times. How much do the experimental
probabilities vary?
the results of 300 simulated throws. Use them
You now have
a head.
to find an experimental probability of throwing
all
d
Experiment with larger numbers of throws, finding an
experimental probability of throwing a head each time.
Comment on your results.

likelihood
probabilities based on equal
can find theoretical
Some situations, you connection between
find experimental probabilities. What is the
d you
end
can also
thetwo?

Summary checklist
to find experimental probabilities
Ican Use the results ot an experiment
them to the theoretical probability.
and compare

287