Probability

Probability

• The probability of an event occurring is between 0 and 1

• If an event is certain not to happen, the probability is 0 eg: the probability of getting a 7 when you roll a die = 0

• If an event is sure to happen, the probability is 1 eg: the probability of getting either a head or a tail when you flip a coin = 1

• All other events have a probability between 0 and 1

Likely or unlikely?

Not likely LikelyImpossible to happen to happen Certain

Very unlikely Equal chance Very Likely to happen of happening to happen

Relative Frequency

This gives information about how often an event occurred compared with other events.

eg: Maths Exam results from 26 students

Exam % No. of students Relative Frequency

> 80 10

60 - 80 12

40 - 60 3

< 40 1

2610

= 0.38 (2 dp)

2612

= 0.46 (2dp)

263

= 0.12 (2 dp)

261

= 0.04 (2 dp)

Sample Space

The set of all possible outcomes is called the sample space.

eg. If a die is rolled the sample space is:{ 1, 2, 3, 4, 5, 6 }

eg. If a coin is flipped the sample space is:{ H, T }

eg. For a 2 child family the sample space is:{ BB, BG, GB, GG }

Equally likely outcomes

In many situations we can assume outcomes are equally likely.

When events are equally likely:

Equally likely outcomes may come from, for example: experiments with coins, dice, spinners and packs of cards

Probability = Number of favourable outcomes

Number of possible outcomes

Pr (getting a 5 when rolling a dice) =61

Favourable Outcomes are the results we wantPossible Outcomes are all the results that are possible - the SAMPLE SPACE

Examples:

Pr (even number on a dice) =21

63

Pr (J, Q, K or Ace in a pack of cards) =5216

= 134

Lattice DiagramsA spinner with numbers 1 to 4 is spun and an unbiased coin is tossed. Graph the sample space and use it to give the probabilities:

(a) P(head and a 4) (b) P(head or a 4)

Sample spaceP(head and a 4) =

81

P(head or a 4) =

(those shaded)

85

Sample Space is {H1, H2, H3, H4, T1, T2, T3, T4}

A spinner with numbers 1 to 4 is spun and an unbiased coin is tossed. We can work out the sample space by a lattice diagram or a tree diagram.

Lattice Diagram

Lattice DiagramsThe sample space when rolling 2 dice can be shown by the following lattice diagram:

1 2 3 4 5 6

1

2

3

4

5

6

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Pr (double) =366

=61

Pr (total ≥ 7) =3621

=127

Tree Diagrams for ProbabilityA tree diagram is a useful way to work out probabilities.

B

B

B

B

BB

B

G

G

G

G

G

G

G

eg: Show the possible combination of genders in a 3 child family

1st child

2nd child

3rd child Outcomes

BBB

BBGBGB

BGGGBB

GBGGGB

GGG

Pr (2 girls & a boy) =

83

Experimental Probability

When we estimate a probability based on an experiment, we call the probability by the term “relative frequency”.

Relative frequency =trialsofnumbertotal

outcomessuccessfulofnumber

The larger the number of trials, the closer the experimental probability (relative frequency) is to the theoretical probability.

Probability Relative Frequency

Theoretical Experimental

Relating factor “is the term we use in”

Pr (getting a 5 when rolling a dice) =61

Favourable Outcomes are the results we wantPossible Outcomes are all the results that are possible – (the sample space)

Examples:

Pr (even number on a dice) =21

63

Pr (J, Q, K or Ace in a pack of cards) =5216

=134

Complementary Events

When rolling a die, ‘getting a 5’ and ‘not getting a 5’ are complementary events. Their probabilities add up to 1.

Pr (getting a 5) =61 Pr (not getting a 5) =

6

5

Using Grids (Lattice Diagrams) to find Probabilities

red die

blue die621

1

2

3

4

5

6

543

●

● ●

●

●●

●●

●●●

●● ●

●

●

●

●

●

● ●

● ●●

●

●

●

●

●

●

●●

●

●

●

●

●

Pr (double) =366

=61

Pr (total ≥ 7) =3621

=127

coin

die654321

H

T ●

●

●

●

●

● ●

●

●

●

●

●

Rolling 2 dice:

Rolling a die & Flipping a coin:

Pr (tail and a 5) =12

1

Pr (tail or a 5) =12

7

Using Grids (Lattice Diagrams) to find Probabilities

1 2 3 4 5 6

1

2

3

4

5

6

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Pr (double) =366

=61

Pr (total ≥ 7) =3621

=127

Multiplying ProbabilitiesIn the previous lattice diagram, when rolling a die and flipping a coin,

coin

die654321

H

T ●

●

●

●

●

● ●

●

●

●

●

●Pr (tail) = Pr (5) =

6

1

2

1

Pr (tail and a 5) =2

1 x6

1=

12

1

So Pr (A and B) = Pr (A) x Pr (B)

example: Jo has probability ¾ of hitting a target, and Ann has probability of ⅓ of hitting a target. If they both fire simultaneously at the target, what is the probability that:

a) they both hit it b) they both miss itie Pr (Jo hits and Ann hits)= Pr (Jo hits) x Pr (Ann hits)= ¾ x ⅓

= ¼

ie Pr (Jo misses and Ann misses)ie Pr (Jo misses) x Pr (Ann misses)= ¼ x ⅔

= 6

1

Tree Diagrams to find ProbabilitiesIn the above example about Jo and Ann hitting targets, we canwork out the probabilities using a tree diagram.

Let H = hit, and M = miss

Jo’s results

H

M

¾

¼

Ann’s results

H

MH

M

⅓

⅔

⅓

⅔

OutcomesH and H

H and MM and H

M and M

Probability¾ x ⅓ = ¼

¾ x ⅔ = ½¼ x ⅓ = 12

1

¼ x ⅔ = 6

1

total = 1Pr (both hit) = ¼

Pr (both miss) = 6

1

Pr (only one hits) ie Pr (Jo or Ann hits) = ½ +12

1= 12

7

●

●or

ExpectationWhen flipping a coin the probability of getting a head is ½, therefore if we flip the coin 100 times we expect to get a head 50 times.

Expected Number = probability of an event occurring x the number of trials

eg: Each time a rugby player kicks for goal he has a ¾ chance of being successful. If, in a particular game, he has 12 kicks for goal, how many goals would you expect him to kick?

Solution: Pr (goal) = ¾

Number of trials = 12

Expected number = ¾ x 12 = 9 goals