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 likely it is, for example, that it will rain tomorrow.

• 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

Probability scale

• 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.

Events and outcomes

• 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

Roulette26

23 1

16

206

12

17

19

14

28

10

30

9

721 3

25

24

13

15

18

27

2

11

29

8

22

4

5

ODD   EVEN

1 11 21

2 12 22

3 13 23

4 14 24

5 15 25

6 16 26

7 17 27

8 18 28

9 19 29

10 20 30

1 to 10 11 to 20 21 to 30

RED   BLACK

P(odd number)

P(1 to 10)

P(Black)

P(number 1)

= 15/30 = ½ or 50%

= 10/30 = 1/3 or 33%

= 15/30 = ½ or 50%

= 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 follows:

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

Number on first dice

Num

ber o

n se

cond

Dic

e

• There are 36 points in this sample space.

• From the sample space we can see, that the sum of 10 occurs three times

• Therefore.....

Other scenarios; Q7

Question 8

1 2 3 4

2 2 3 8

3 4 6 12

Experimental probability

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

Theoretical probability can be found without doing and experiment.

EXPERIMENTAL

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

THEORETICAL PROBABILITY

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

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

HEADS

TAILS

THEORETICAL PROBABILITY

• When flipping the coin, the probability that my coin will land on heads is 1 in 2

• What is the probability that my coin will land on tails??

HEADS

TAILS

Theoretical Probability

HEADS

TAILS

A probability of 1 in 2 can be written in two ways:

•As a fraction: ½

•As a decimal: .50

Theoretical probability

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

A

A

A

A

A 1 in 4 chance can be written 2 ways:

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

A

A

A

Theoretical Probability

• I am going to take 1 marble from the bag.• What is the probability that I will pick out

a red marble?

Theoretical Probability

I have three marbles in a bag.

1 marble is red

1 marble is blue

1 marble is green

Theoretical Probability• 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.

Experimental Probability

• 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.

Experimental Probability

• We draw 1 marble from the bag.

Marble number red blue green

1 123456

It is a red marble.

Record the outcome on the tally sheet

Experimental Probability

• 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 number red blue green

1 12 134

Experimental Probability

• 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)

Experimental Probability

• After 6 draws your chart will look similar to this.

• Look at the red column.• Of our 6 draws, we

selected a red marble 2 times.

Marble number red blue green

1 12 13 14 15 16 1

Total 2 1 3

Experimental Probability

• The experimental probability of drawing a red marble was 2 in 6.

• This can be expressed as a fraction: 2/6 or 1/3 a decimal : .33 or a percentage: 33%

Marble number red blue green

1 12 13 14 15 16 1

Total 2 1 3

Experimental Probability

• Notice the Experimental Probability of drawing a red, blue or green marble.

Marble number red blue green

1 12 13 14 15 16 1

Total 2 1 3

Exp. Prob.

2/6 or 1/3 1/6

3/6 or 1/2

Comparing Experimental and Theoretical Probability

• Look at the chart at the right.

• Is the experimental probability always the same as the theoretical probability?

red blue greenExp. Prob. 1/3 1/6 1/2Theo. Prob. 1/3 1/3 1/3

Comparing Experimental and Theoretical Probability

• In this experiment, the experimental and theoretical probabilities of selecting a red marble are equal.

red blue greenExp. Prob. 1/3 1/6 1/2Theo. Prob. 1/3 1/3 1/3

Comparing Experimental and Theoretical Probability

• The experimental probability of selecting a blue marble is less than the theoretical probability.

• The experimental probability of selecting a green marble is greater than the theoretical probability.

red blue greenExp. Prob. 1/3 1/6 1/2Theo. Prob. 1/3 1/3 1/3

Probability Review

• Theoretical (can be found without doing an experiment)

• Experimental (can be found by repeating an experiment and recording outcomes.)

There are 2 types of probability:

Probability is a number from 0 to 1 that tells you how likely something is to happen.

Questions 4.4

Mutually exclusive events:4.5

• Question 1:• Unbiased dice results:

1 23 4 5 6

Question 3

Question 4