Probability & chance

• If you listen to weather forecasts you could
hear expressions like these:

• ‘There is a strong likelihood of rain
tomorrow’.

• ‘In the afternoon there is a possibility of
thunder’.

• ‘The rain will probably clear towards
evening’.

• Weather forecasts are made by studying
charts and weather data to tell us how

• Probability uses numbers to tell us
how likely something is to happen.

• The probability or chance of
something happening can be
described by using words such as

• Impossible, Unlikely, Even, Chance,
Likely or Certain

• An event which is certain to happen
has a probability of 1.

• An event which cannot happen has a
probability of 0.

• All other probabilities will be a
number greater than 0 and less than 1.

• The more likely an event is to happen,
the closer the probability is to 1

• There is an even chance that the
next person you meet on the Street
will be a male.

• It is certain that the sun will rise
tomorrow.

• It is impossible to get 7 when a
normal dice is rolled.

• Before you start a certain game you
must throw a dice and get a six
• The act of throwing is called a
trial
• The numbers 1,2,3,4,5,6 are the
possible outcomes
• The required result is called the
event

• In general the letter E represents
the event, probability is denoted by
the letter P
• The formal definition of
probability is as follows

• The probability of any event cannot
be less than 0 or greater than 1
• The probability of a certainty is 1
• An impossibility is 0

Example 1
• A card is drawn from a pack of 52
playing cards. Find the probability
that the card is (i)a diamond (ii) a
queen (iii) a king or a queen
• (i)There are 13 diamonds in a pack
therefore

• (ii) there are 4 queens in a pack
therefore:

• (iii) there are 8 queens or kings in a
pack therefore

Roulette
ODD EVEN

2 2 1 11 21
2 1
6 6 3 1
0 2 12 22
1 6 5
3 13 23
2
1 4
4 14 24
7
1 2 5 15 25
9 2
1 8 6 16 26
4 7 17 27
2 2
8 8 18 28
9
1 1 9 19 29
0 1
10 20 30
3 2
0 1 to 10 11 to 20 21 to 30
9 2 RED BLACK
7
7 1
2 8
1
1 3
2 2
1
3
5 P(odd number) = 15/30 = ½ or 50%
5 4
P(1 to 10) = 10/30 = 1/3 or 33%
P(Black) = 15/30 = ½ or 50%
P(number 1) = 1/30 or 3.3%

Probability of an event not
occurring
• The probability of drawing spade
from a pack of cards is....

• Therefore the probability of not
drawing a spade is simply the
probability of drawing any other
card in the pack, therefore...

• This illustrates the probability of not
drawing a spade is one minus the
probability of drawing a spade ,
written as...

Two events –the use of
sample space
• When two coins are tossed the set
of possible outcomes is as follows
• There could be two heads
• There could be a head and a tail
• There could be a tail and a head
• Or there could be two tails

• This is written as follows:
• {HH,HT,TH,TT}
• Where H=head and T=tail
• This set of possible outcomes is
called sample space. By using this
sample space we can write down
the probability of { HH } for
example as

• The probability of one head and one
tail is obtained by taking HT and TH

• Similarly if two dice are thrown
and the numbers on the dice are
added, we can set out sample space
of results as Number on first dice
follows:
1 2 3 4 5 6
Number on second Dice

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

• There are 36 points in this sample
space.
• From the sample space we can see,
that the sum of 10 occurs three
times
• Therefore.....

Question 8

1 2 3 4

2 2 3 8

3 4 6 12

What is Probability?

• Probability is a number from 0 to 1 that tells
you how likely something is to happen.
• Probability can be either theoretical or
experimental.

Probability

THEORETICAL EXPERIMENTAL

Theoretical probability Experimental probability is
can be found without found by repeating an
doing and experiment and
experiment. observing the
outcomes.

THEORETICAL PROBABILITY

• Take for example a coin
It has a heads side and a
tails side HEADS

 Since the coin has only 2
sides, there are only 2
possible outcomes when TAILS
you flip it. It will either
land on heads, or tails

THEORETICAL PROBABILITY

• When flipping the coin,
the probability that my HEADS
coin will land on heads is
1 in 2
• What is the probability
TAILS
that my coin will land on
tails??

Theoretical Probability

A probability of 1 in 2 can
be written in two ways:
•As a fraction: ½ HEADS

•As a decimal: .50

TAILS

Theoretical probability

When I spin this
spinner, I have a 1 in
4 chance of landing A A
on the section with
the red A in it. A A

A 1 in 4 chance can be written 2 ways:

• As a fraction: ¼
• As a decimal: .25 A A
A A

I have three marbles in a bag.
1 marble is red
1 marble is blue
1 marble is green
• I am going to take 1 marble from the bag.
• What is the probability that I will pick out
a red marble?

• Since there are three
marbles and only one is
red, I have a 1 in 3 chance
of picking out a red
marble.
• I can write this in two
ways:
• As a fraction: 1/3
• As a decimal: .33

Experimental Probability

Experimental probability is found
by repeating an experiment and
observing the outcomes.


• Returning again to the bag of
marbles?
• The bag has only 1 red, 1 green, and 1
blue marble in it.
• There are a total of 3 marbles in the
bag.
• Theoretical Probability says there is a
1 in 3 chance of selecting a red, a
green or a blue marble.

• We draw 1 marble from the bag.

 It is a red marble.

Record the outcome on the tally sheet
Marble
number red blue green
1 1
2
3
4
5
6


• If we put the red marble back in the bag
and draw again.
• This time you drew a green marble.
• Record this outcome on the tally sheet.
Marble
1 1
2 1
3
4

• We place the green marble back in the bag.
• We then continue drawing marbles and
recording outcomes until we have drawn 6
times. (remember it is essential that each
marble is placed is back in the bag before
drawing again)


• After 6 draws your chart
Marble
will look similar to this. number red blue green
• Look at the red column. 1 1
2 1
• Of our 6 draws, we 3 1
selected a red marble 2 4 1
5 1
times. 6 1
Total 2 1 3


• The experimental
Marble
probability of drawing a
red marble was 2 in 6. 1 1
• This can be expressed as a 2 1
fraction: 2/6 or 1/3 3 1
4 1
a decimal : .33 or
5 1
a percentage: 33% 6 1
Total 2 1 3


Marble
• Notice the number red blue green
Experimental 1 1
2 1
Probability of
3 1
drawing a red, 4 1
blue or green 5 1
6 1
marble. Total 2 1 3
2/6 3/6
Exp. or or
Prob. 1/3 1/6 1/2

Comparing Experimental and
• Look at the chart at
the right.
• Is the experimental red blue green
Exp.
probability always the
Prob. 1/3 1/6 1/2
same as the Theo.
theoretical Prob. 1/3 1/3 1/3
probability?

• In this experiment, the
experimental and
red blue green
theoretical
Exp.
probabilities of
Prob. 1/3 1/6 1/2
selecting a red marble Theo.
are equal. Prob. 1/3 1/3 1/3

• The experimental
probability of selecting a
blue marble is less than the
red blue green
theoretical probability.
Exp.
• The experimental Prob. 1/3 1/6 1/2
probability of selecting a Theo.
green marble is greater Prob. 1/3 1/3 1/3
than the theoretical
probability.

Probability Review

Probability is a number from 0 to 1 that tells
you how likely something is to happen.
There are 2 types of probability:
• Theoretical (can be found without doing an
experiment)
• Experimental (can be found by repeating an
experiment and recording outcomes.)

Mutually exclusive events:4.5
• Question 1:
• Unbiased dice results:

1 2 3 4 5 6

Probability & chance

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