Probability (1)Probability (1)

Outcomes and EventsOutcomes and Events

Let C mean “the Event a Court Card (Jack, Queen, King) is chosen”

Let D mean “the Event a Diamond is chosen”

Probability Notation

n(C) means “the number of outcomes favourable to C”

n(D) means “the number of outcomes favourable to D”

n(C) = 12 (4x3=12 Court cards in a pack of 52)

n(D) = 13 (13 Diamonds in a pack of 52)

Venn Diagram

CThe court

cards

DDiamonds

Outside is all other cards

Cards that are Court Cards and Diamond

C D

Probability Notation (2)

n(C) = 12 (12 Court cards in a pack of 52)

n(D) = 13 (13 Diamonds in a pack of 52)

Let C mean “the Event a Court Card is chosen”Let D mean “the Event a Diamond is chosen”

C D means “the Event a card that is both a court card and diamond is chosen”

n(C D) = 3 (the Jack, Queen, King Diamonds)

n(C n D) means “the number of outcomes of both events C and D”

Fish ‘n’ Chips

Fish Chips

Venn Diagram

F CF n C

Venn Diagram

CThe court

cards

DDiamonds

Outside is all other cards

C D

3

12 13

30 ?

?

?

?

Venn Diagram

C D

C DEntire Shaded area is the ‘Union’

Venn Diagram

C D

C Dn(C)=12

n(C D) = 3

n(D)=13

n(C D) = n(C) + n(D) - n(C D)

12n(C D) =

Avoid double-counting these

+ 13 - 3 = 22

Probability Notation (3)Let C mean “the Event a Court Card is chosen”Let D mean “the Event a Diamond is chosen”

C D means “the Event a card that chosen is a court card or a diamond”

n(C u D) means “the number of outcomes of C or D”

n(C D) = n(C) + n(D) - n(C D)

Venn Diagram

C

n(C) = 12

n(C’) = 40

C’The complement

?

Cn(C) = 12

P(C) The probability

of C

= n(C) = 12 = 3 52 52 13

C’

P(C’)

n(C’) = 40

= n(C’) = 40 = 10 52 52 13

P(C’) = 1 - P(C)

Venn Diagram

C DC DP(C) = n(C)/52 P(D) =

n(D)/52P(CnD) = n(CnD)/52

P(CnD) = n(CnD)/52 = 3/52“The probability of choosing a card that is

both a Court Card and a Diamond is 3/52”

___52

___52

______52

______52

Venn Diagram

C DC DP(C) = n(C)/52

n(C D) = n(C) + n(D) - n(C D)

P(D) = n(D)/52

P(CnD) = n(CnD)/52

P(C D) = P(C) + P(D) - P(C D)

Venn Diagram

C D

If there is no overlap, it means there are no outcomes in common

n(C D) = 0

These are known as MUTUALLY EXCLUSIVE EVENTS

For example:- C means “picking a Court Card” D means “picking a Seven”

Probability Notation (4)

P(C) The probability

of C

= n(C) 52

P(C’) = 1 - P(C)

P(C D) = P(C) + P(D) - P(C D)

P(C D) = P(C) + P(D)

For mutually exclusive events

n(C D) = n(C) + n(D)

Consider a series of 60 Maths Lessons

If ...

P(L G) = ?

P(L) = ?

P(G) = ?

P(L G) = ?

Lisa is absent 40 times

Gus is absent 18 times

In 5 lessons they were both absent

What does mean (in words) ?P(L G)

5/60 = 1/12

18/60 = 3/10

40/60 = 2/3

L G

L GP(L G) = 1/12

P(L)=2/3 P(G)=3/10

P(L G) = P(L) - P(L G)+ P(G)

2/3P(L G) = + 3/10 - 1/12 = 53/60“In 53/60 lessons Lisa or Gus was absent”

they are both absent

Consider a series of 60 Maths LessonsIf ...

P(L) = 2/3

P(G) = 3/10

Lisa absent the most

P(L G) = 1/12

P(L G) = 53/60

Gus absent the most

Both absent

Either is absent

P(L’) = ?

What is the probability Lisa isn’t absent?

P(L’) = 1 - P(L) = 1 - 2 = 1 3 3