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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

Not likely Likely

Impossible to happento happen Certain

Very unlikelyEqual chanceVery Likely

to happenof happening to happen

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

eg: Maths Exam results from 26 students

= 0.38 (2 dp)

= 0.46 (2dp)

= 0.12 (2 dp)

= 0.04 (2 dp)

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

{ 1, 2, 3, 4, 5, 6 }

eg. If a die is rolled the sample space is:

{ H, T }

eg. If a coin is flipped the sample space is:

{ BB, BG, GB, GG }

eg. For a 2 child family the sample space is:

In many situations we can assume outcomes are equally likely.

When events are equally likely:

Probability = Number of favourable outcomes

Number of possible outcomes

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

Favourable Outcomes are the results we want

Possible Outcomes are all the results that are possible - the SAMPLE SPACE

Examples:

Pr (getting a 5 when rolling a dice) =

Pr (even number on a dice) =

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

=

Sample space

P(head and a 4) =

P(head or a 4) =

(those shaded)

A 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)

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

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

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

1 2 3 4 5 6

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

1

2

3

4

5

6

Pr (double) =

=

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

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

Pr (total ≥ 7) =

=

(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)

A tree diagram is a useful way to work out probabilities.

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

Pr (2 girls & a boy) =

3rd child

Outcomes

2nd child

B

BBB

B

1st child

G

BBG

B

B

BGB

G

BGG

G

GBB

B

B

G

GBG

G

GGB

B

G

G

GGG

Probability Relative Frequency

Theoretical Experimental

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

Relative frequency =

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

Relating factor “is the term we use in”

Favourable Outcomes are the results we want

Possible Outcomes are all the results that are possible – (the sample space)

Examples:

Pr (getting a 5 when rolling a dice) =

Pr (even number on a dice) =

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

=

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) =

Pr (not getting a 5) =

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6

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5

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4

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red die

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3

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2

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1

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5

1

2

3

4

6

blue die

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T

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coin

H

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6

5

1

2

3

4

die

Using Grids (Lattice Diagrams) to find Probabilities

Rolling 2 dice:

Pr (double) =

=

Pr (total ≥ 7) =

=

Rolling a die & Flipping a coin:

Pr (tail and a 5) =

Pr (tail or a 5) =

Using Grids (Lattice Diagrams) to find Probabilities

1 2 3 4 5 6

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

1

2

3

4

5

6

Pr (double) =

=

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

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

Pr (total ≥ 7) =

=

(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)

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T

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coin

H

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6

5

1

2

3

4

die

Multiplying Probabilities

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

Pr (tail) =

Pr (5) =

Pr (tail and a 5) =

x

=

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 it

ie Pr (Jo hits and Ann hits)

ie Pr (Jo misses and Ann misses)

= Pr (Jo hits) x Pr (Ann hits)

ie Pr (Jo misses) x Pr (Ann misses)

= ¾ x ⅓

= ¼ x⅔

= ¼

=

Tree Diagrams to find Probabilities

In the above example about Jo and Ann hitting targets, we can

work out the probabilities using a tree diagram.

Let H = hit, and M = miss

Ann’s results

Probability

Outcomes

H and H

¾ x ⅓ = ¼

⅓

H

Jo’s results

H

¾

⅔

¾ x ⅔ = ½

H and M

●

M

or

M and H

¼ x ⅓ =

H

●

⅓

¼

M

⅔

M and M

¼ x ⅔ =

M

Pr (both hit) = ¼

total = 1

Pr (both miss) =

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

=

Expectation

When 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