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Introduction

Cards

Combining Probabilities

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IntroductionIntroduction

• Probability is the maths of chance and gambling, telling us how likely an event is to occur.

• The probability of an event occurring is usually written as a fraction or as a percentage.

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• Toss one coin, assume it is a fair coin and ignore the chance of it landing on its edge.

P(Head) =12__

TH

• There are only two possible outcomes – either a head H or a tail T.

P(Tail) =12__

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DefinitionDefinitionThe probability of an event occurring is:

P (event) = Total possible number of outcomes

Number of times this event occurs–––––––––––––––––––––––––––––

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• Consider a deck of cards. There are four suits called hearts, diamonds, clubs and spades.

Cards Cards

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• Consider a deck of cards. There are four suits called hearts, diamonds, clubs and spades.

• In each of these there are 13 cards – an ace, the numbers 2 to 10 inclusive, and the picture cards: jack, queen and king.

• This makes a total of 52 cards in an ordinary deck. Some games require one extra card, the Joker. We will not use this extra card.

• If the cards are boxed (shuffled or mixed) and then 13 cards dealt to each of four people, the chances of a particular person getting 13 clubs are 635,013,559,600 to 1.

• This is more than the number of seconds in 20,000 years!

Cards Cards

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Clubs A 2 3 4 5 6 7 8 9 10 J Q K

Diamonds A 2 3 4 5 6 7 8 9 10 J Q K

Hearts A 2 3 4 5 6 7 8 9 10 J Q K

Spades A 2 3 4 5 6 7 8 9 10 J Q K

P(Clubs) =1352___

Cards Cards

number of clubs in decktotal number of cards =

14__

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Clubs A 2 3 4 5 6 7 8 9 10 J Q K

Diamonds A 2 3 4 5 6 7 8 9 10 J Q K

Hearts A 2 3 4 5 6 7 8 9 10 J Q K

Spades A 2 3 4 5 6 7 8 9 10 J Q K

P(2) = 452___

Cards Cards

number of 2s in decktotal number of cards =

113__

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Combining ProbabilitiesCombining Probabilities

• If two events happen simultaneously, a sample space can be constructed to see clearly the possible outcomes.

• A sample space involves putting all the possible outcomes of one event on one axis of a grid and all the possible outcomes of a second event on the other axis.

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1 2 3 4 5 6

1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6

2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6

3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6

4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6

5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6

6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6

DiceDiceThrow 2

Throw 1

36

P(both same) = 636___ number the same

total outcomes = 1 6__

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1 2 3 4 5 6

1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6

2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6

3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6

4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6

5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6

6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6

DiceDiceThrow 2

Throw 1

36

P(two 4s) = 136___ number the same

total outcomes = 136__

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1 2 3 4 5 6

1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6

2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6

3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6

4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6

5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6

6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6

DiceDiceThrow 2

Throw 1

36

P(at least one 6) =1136___

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1 2 3 4 5 6

1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6

2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6

3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6

4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6

5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6

6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6

DiceDiceThrow 2

Throw 1

36

P(not getting a 6) =2536___

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1 2 3 4 5 6

1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6

2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6

3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6

4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6

5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6

6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6

DiceDiceThrow 2

Throw 1

36

P(a total of 10) = 336___ =

112__

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Three or more eventsThree or more events

T

H

T

H

T

H

T

H

H

T

H

T

T

H

HHH

HHT

HTH

HTT

THH

THT

TTH

TTT

P(2 heads and 1 tail) = 3 8___

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