Probability NCEA Level 1. There are 32 students in this class. 15 have seen “the Dark Knight”. 12 want to see the movie but haven’t. 5 don’t want to see.

ProbabilityProbability

NCEA Level 1NCEA Level 1

There are 32 students in this class. 15 have There are 32 students in this class. 15 have seen “the Dark Knight”. 12 want to see the seen “the Dark Knight”. 12 want to see the movie but haven’t. 5 don’t want to see the movie but haven’t. 5 don’t want to see the movie. If one random student is selected. movie. If one random student is selected. What’s the probability he or she wants to What’s the probability he or she wants to watch the movie?watch the movie?

DefinitionsDefinitions experimentsexperiments : :Corne and Blake Corne and Blake toss a cointoss a coin to decide who get to go to the to decide who get to go to the

bathroom firstbathroom firstRya Rya draws a Tarot carddraws a Tarot card to predict her future to predict her future

TrialTrial One performance of the experiment:One performance of the experiment: Corne and Blake Corne and Blake toss a coin 3 times ( 3 trials)toss a coin 3 times ( 3 trials)Madison has bought lotto ticket Madison has bought lotto ticket 30 times (30 trials)30 times (30 trials)

outcomeoutcome The result of a trial of an experimentThe result of a trial of an experiment Rya Rya draws out a “knightdraws out a “knight” from her tarot set.” from her tarot set. The coin landed on The coin landed on head.head. (Blake gets to go first)(Blake gets to go first)

Sample spaceSample space The set of all the possible outcomes of an The set of all the possible outcomes of an

experiment.experiment. Tossing a coin—Head and tailTossing a coin—Head and tail Drawing a Tarot card –The sun, the moon, the knight, the Drawing a Tarot card –The sun, the moon, the knight, the

queen, the king……a total of 25!queen, the king……a total of 25!

EventEvent The combination of one or more outcomesThe combination of one or more outcomes Rya gets both a ‘knight’ and a “prince”Rya gets both a ‘knight’ and a “prince”

FrequencyFrequency The number of times the event occursThe number of times the event occurs The number of time Madison wins lotto (0)The number of time Madison wins lotto (0) The number of times the coin lands on head. The number of times the coin lands on head.

ProbabilityProbabilityIs expressed as Is expressed as fraction or percentage.fraction or percentage.

P(E) = P(E) =

Time Time spent on spent on maths maths (hours)(hours)

FrequencyFrequency

0.1-0.250.1-0.25 33

0.25-0.5 0.25-0.5 1515

0.5-1 0.5-1 77

1-1.51-1.5 33

1.5-2 1.5-2 22

P( 1.5-2 hours)= P( 1.5-2 hours)=

P(less than 0,5)=P(less than 0,5)=

P(less than 0.1)=P(less than 0.1)=

P(more than 0.1)=P(more than 0.1)=

Ms Li’s 11 Mat class has been surveyedOn how many hours they spend on maths Per day. There are 32 students in her class.

Real lifeReal life In many real life situations we In many real life situations we

can only estimate the can only estimate the probability of an event. We do probability of an event. We do this by collecting data and this by collecting data and working out proportions.working out proportions.

Example:Example: The table shows the type of The table shows the type of

books on a bookshelf.books on a bookshelf.

Hard Hard backback

Paper Paper backback

totaltotal

Non-Non-fictionfiction

66 44 1010

fictionfiction 22 2020 2222

totaltotal 88 2424 3232

Hard Hard backback

Paper Paper backback

totaltotal

Non-Non-fictionfiction

66 44 1010

fictionfiction 22 2020 2222

totaltotal 88 2424 3232

(a)If one book is taken off the shelf at random, calculate the

P(paperback, non-fiction)

P(fiction)

Jack has been Jack has been keeping records of his keeping records of his wins and losses for wins and losses for the different kinds of the different kinds of games he plays, in games he plays, in order to find out in order to find out in which games he has which games he has most success. His most success. His tabulated records are tabulated records are shown below:shown below:

WinsWins LossesLosses TotalTotal

Ball gamesBall games 130130 7070 200200

Card gamesCard games 141141 5858 199199

Board Board gamesgames

100100 7575 175175

TotalTotal 371371 203203 574574

WinsWins LossesLosses TotalTotal

Ball Ball gamesgames

130130 7070 200200

Card Card gamesgames

141141 5858 199199

Board Board gamesgames

100100 7575 175175

TotalTotal 371371 203203 574574

a. What is the probability that Jack wins a board game?

b. What is the probability that the last game Jake won was a board game?

16

16

32

8 8

8

8

8

8

8

8

16 16

1616

1616

Probability TreeProbability Tree

A bag contains three red balls, four blue A bag contains three red balls, four blue balls and five green balls. If two balls are balls and five green balls. If two balls are drawn, with replacement, draw a drawn, with replacement, draw a probability tree to find out the probability probability tree to find out the probability that:that:

a. The first ball will be green?a. The first ball will be green?b. The first ball is red and the second ball b. The first ball is red and the second ball is blue? is blue?c. Both balls are red?c. Both balls are red?d. Both balls are not green?d. Both balls are not green?

Tree DiagramsTree Diagrams School is selling ‘scratch & School is selling ‘scratch &

win’ cards as a fundraising win’ cards as a fundraising project. Each card has a project. Each card has a chance of 1/3 of winning a chance of 1/3 of winning a prize. Hoony has bought 2 prize. Hoony has bought 2 cards.cards.

(a) Show the outcomes in a (a) Show the outcomes in a tree diagram each with their tree diagram each with their probability of happening.probability of happening.

(b) Calculate P(one prize)(b) Calculate P(one prize) (c) Calculate P(at least one (c) Calculate P(at least one

prize)prize)

Mikhail and Robert play two games of tennis. The probability of Mikhail winning the first game is 1/3 . If he wins the first

game, the probability of him winning the second game is 2/3, but if he loses the first game the probability of him losing the

second game is 1/2.

(a)Draw a probability tree for these two games(b)What is the probability of Mikhail winning both games?(c)What is the probability of Mikhail losing both games?(d)What is the probability of Robert winning one game and losing

one game?

Sampling without replacementSampling without replacement

In this type of probability tree we DO NOT put the object back so there are fewer to select from the next time round.

Each time we select a choice it is then gone and the subsequent number of options is reduced.

Probabilities on following branches are changed.

R

G

B

R

G

B

R

G

B

Eg marbles in a bag 3 red, 2blue, 5 green. We take out one and then put it back and then take out another and put it back.

R

G

B

Out comesRR

RB

RGBR

BB

BGGR

GB

GG 0.5 0.5

0.3 0.3

0.3 0.2

0.3 0.50.2 0.3

0.2 0.2

0.2 0.50.5 0.3

0.5 0.2

R

G

B

R

G

B

R

G

B

Eg marbles in a bag 3 red, 2blue, 5 green. We take out one and then DON’T put it back and then take out another.

R

G

B

RR

RB

RGBR

BB

BGGR

GB

GG5 4

10 9

3 2

10 9

3 2

10 9

3 5

10 9

2 3

10 9

2 1

10 9

2 5

10 9

5 3

10 9

5 2

10 9

What is the probability of getting a green and a blue marble?What is the probability of getting two marbles the same colour?

Jaz goes to Rome. She has a guide book which lists 19 important statues. 9 were erected in the 19th century and 10 were erected in the 20th century. If she visits two statues at random what is the probability that both were erected in the 19th century

19th

20th

19th

19th

20th

20th

9

19

10

19

8

18

10

18

9

18

9

18

P(19,19)

9 8=

19 184

19

YearYear BrownBrown

HairHair

Black Black HairHair

OtherOther TotalTotal

19981998 450450 200200 120120

19991999 425425 250250 100100 775775

20002000 425425 275275 8585

20012001 400400 200200 100100 700700

20022002 375375 225225 125125

TotalTotal 11501150

770

785

725

2075 530 3755

400)

700a 4

7

425 85)

785b

510

785

102

157

2075)

3755c

85

151

What is the probability that a person enrolling at Last Chance College:

a) In 2001, will have brown hair? (conditional probability)b) In 2000, will not have black hair? (conditional probability)c) In the 5 year period would have had brown hair?

TallTall DwarfDwarf TotalTotal

RedRed 2323 2222 4545

BlueBlue 4545 1515 6060

YellowYellow 3737 2828 6565

105105 6565 170170

A scientist plants seeds. When the seeds came up he recorded the results.

h) A blue flower is

dwarf

(conditional)

i) A yellow flower is

tall (conditional)

j) A flower is yellow

15( )

601

4

P BD

37( )

65P YT

65( )

17013

34

P Y

Conditional Conditional ProbabilityProbability

Conditional ProbabilityConditional Probability

0.482) Given that the first marble drawn out was green, what is the probability that the second marble is red?Given the first marble was green, tells us that we can miss out the first part of tree and start at the second branch.

If you already know some conditions then it eliminates part of the tree so we only need to consider the relevant parts

The probability of the second marble being red is 0.6

There are 6 red marbles and 4 green marbles in a bag.1) What is the probability of drawing out a red and a green marble?

0.6

0.4

R

G

R

R

G

G

0.6

0.4

0.6

0.4

P(R & G) P(RG) or P(GR)0.6 0.4 0.4 0.6

0.24 0.24

Students were asked if they had takeaways in the last week45% of the students were male. if they were male the probability they had takeaways was 40%. If they were female the probability that they had takeaways was 30% If the females had takeaways there was a 25% chance they had fish and chips. If they were male there was a 35% chance they had fish and chips1) Draw a probability tree to represent this situation

M

F

T

NT

T

NT

F&C

O

F&C

O

0.45

0.55

0.4

0.6

0.3

0.7

0.35

0.65

0.25

0.75

2) What is the probability that a student chosen at random has fish and chips?

0.10425

M

F

T

NT

T

NT

F&C

O

F&C

O

0.45

0.55

0.4

0.6

0.3

0.7

0.35

0.65

0.25

0.75

P(M&T&FC) or P(F&T&FC)(0.45 0.4 0.35) (0.55 0.3 0.25) 0.063 0.04125

3) Given that a student is female what is the probability that they has fish and chips?As it is given they are female we start after that branch

0.3 0.25 0.075

M

F

T

NT

T

NT

F&C

O

F&C

O

0.45

0.55

0.4

0.6

0.3

0.7

0.35

0.65

0.25

0.75

( & )P T FC

The most common campervan booking at SeeNZ is for a 10-day length of time.85% of customer requests for a 10-day booking can be confirmed immediately.The rest of the customer requests for a 10-day booking go on a waiting list.Only 25% of those on the waiting list eventually have their request for a 10-day bookingconfirmed.

SeeNZ had to turn away 20 people during a three month period, because their request for a 10-day booking could not be confirmed.

Calculate the total number of customer requests for a 10-day booking SeeNZ received duringthat three month period.

2007 Excellence Question

177

PROBABILITYPROBABILITYSIMULATIONSSIMULATIONS

This is when we construct an This is when we construct an experiment to give us an idea of what experiment to give us an idea of what might happen in a particular situation.might happen in a particular situation.

Image if Robert and Murphy play table tennis. Image if Robert and Murphy play table tennis.

There is 50% and 50% chance of winning for There is 50% and 50% chance of winning for both of them. How would you simulate the game both of them. How would you simulate the game 10 times? What tool would you use to help you?10 times? What tool would you use to help you?

One out of Six people will be randomly chosen One out of Six people will be randomly chosen to go on a field trip. How would you simulate the to go on a field trip. How would you simulate the choosing process?choosing process?

Every year Nick, Matt and Charlotte compete for Every year Nick, Matt and Charlotte compete for a maths scholarship. Nick and Matt both have a maths scholarship. Nick and Matt both have 25% chance of getting the scholarship. Charlotte 25% chance of getting the scholarship. Charlotte has 50% chance of getting the scholarship. How has 50% chance of getting the scholarship. How would you simulate the situation for 6 years?would you simulate the situation for 6 years?

Key ConceptsKey Concepts TToolsools What are you using – die, cards, spinner, calculator (Ran#)What are you using – die, cards, spinner, calculator (Ran#)

TTrialsrials Define each trialDefine each trial How many trials will be neededHow many trials will be needed

LList (Results)ist (Results) List the results in tablesList the results in tables

CCalculationalculation Answer questions by calculating the probabilityAnswer questions by calculating the probability

Simulation QuestionSimulation Question

There are three sets of lights along Maths There are three sets of lights along Maths Highway. Each light is red 30% of the Highway. Each light is red 30% of the time. time.

Design and carry out a simulation to Design and carry out a simulation to predict how many cars out of 50 would go predict how many cars out of 50 would go along the highway without stopping at any along the highway without stopping at any lights.lights.

There are three sets of lights along Maths There are three sets of lights along Maths Highway. Each light is red 30% of the Highway. Each light is red 30% of the time. time.

Design and carry out a simulation to Design and carry out a simulation to predict how many cars out of 50 would go predict how many cars out of 50 would go along the highway without stopping at any along the highway without stopping at any lights.lights.

TOOLTOOL Use a calculator to create random Use a calculator to create random

numbers from 1 to 10numbers from 1 to 10 with 1, 2, 3 representing -Stop (Red) with 1, 2, 3 representing -Stop (Red)

and 4, 5, 6, 7, 8, 9, 0 representing Go and 4, 5, 6, 7, 8, 9, 0 representing Go (Green) (Green)

TrialTrial

Select 3 random numbers and record Select 3 random numbers and record them. Each number represent a them. Each number represent a different colour light.different colour light.

If all 3 are GO put a tick in the result If all 3 are GO put a tick in the result column. Repeat the trial column. Repeat the trial 5050 times. times.

TrialTrial    OUTCOMEOUTCOME    LIGHTLIGHT ResultResult

   1st#1st# 2nd#2nd# 3rd#3rd# 1st1st 2nd2nd 3rd3rd   

11 11 66 33 stopstop GoGo stopstop

22 77 44 88

GoGo GoGo GoGo

33

33 00 11

stopstop GoGo stopstop

44                     

55                     

66                     

77                     

88                     

99                     

  5050                     

Results

CALCULATIONCALCULATION

P(Not Stopping)=P(Not Stopping)= No ofNo of

5050

Madison wants to go on a cruise with 6 of her friends.

There is a special discount for groups of 5. She can go if she can get 4 other friends to go. There is a 50% chance that each of her friends will go.

a) Find and estimate of the probability that she will have at least 4 of her friends will go

b) What is the most likely number of friends that will want to go?

1) Tool:  Flip a coin : Heads go, Tails not go 2) Trial: Flip a coin 6 times- one trial Perform 50 trials3) Results 11 22 33 44 55 66 No. No.

that gothat goAt least 4 At least 4

gogo

HH HH TT TT HH HH 44 YY

TT HH TT HH TT HH 33 NN

HH HH HH HH HH TT 55 YY

etcetc

4. Calculationa)no. of Y 50b) The most frequently occurring

number of friends is the most likely number that will want to go

Example 2Example 2

Five cards are randomly inserted into cereal Five cards are randomly inserted into cereal packets. They each carry one of the letters P, R, I, packets. They each carry one of the letters P, R, I, Z or E. What is the probability that all five cards are Z or E. What is the probability that all five cards are obtained if 10 cereal boxes are bought.obtained if 10 cereal boxes are bought.

A TOOL is needed to simulate the situationA TOOL is needed to simulate the situation A TRIAL needs to be describedA TRIAL needs to be described LIST. How will data be recorded?LIST. How will data be recorded? CALCULATION p(all 5 cards obtained) = no. CALCULATION p(all 5 cards obtained) = no.

successes/30successes/30

A dog breeder wants to know the average number of puppies produced by his 600 dogs over a period of three breeding seasons. Assume that each dog produces either a single puppy or twins. From the record of past breeding seasons the breeder know that the probability of having twin puppies is 1/5.

Design a model to simulate the breeding of puppies over three successive breeding seasons.

Use the results of your simulations to find the mean number of puppies produced by a dog over the three seasons.

The breeder decided if the dog produced two sets of twins will no longer be used for breeding purposes. Use theoretical probability to find how many of the 600 dogs will no longer be suitable for breeding after three years.

Simulation guard and thiefSimulation guard and thief

The Guard and the ThiefThe Guard and the ThiefAn office building has a cash register and a An office building has a cash register and a

safe in two different places. The register safe in two different places. The register has $2000, the safe has $10,000.has $2000, the safe has $10,000.

Every night, there is a guard and a thief. The Every night, there is a guard and a thief. The guard spends 2/3 of the time by the safe, guard spends 2/3 of the time by the safe, 1/3 of the time at the cash register. 1/3 of the time at the cash register.

The thief only goes to the cash register 1/6 The thief only goes to the cash register 1/6 of the time. 5/6 of the time he goes to the of the time. 5/6 of the time he goes to the safe.safe.

At the end of ten days, how much money At the end of ten days, how much money will the thief have stolen? (Assume that he will the thief have stolen? (Assume that he doesn’t get caught in these 10 days)doesn’t get caught in these 10 days)

Tool:Tool:Use a dice. Use a dice.

TrailTrail One trail is one throw of One trail is one throw of

the dice.the dice. Guard:Guard: at cash register: at cash register: throw 1 and 2throw 1 and 2 at safe: at safe: throw 3, 4, 5 and 6throw 3, 4, 5 and 6 Thief:Thief: at cash register: at cash register: throw 1throw 1 at safe: at safe: Throw 2,3,4,5,6Throw 2,3,4,5,6

ToolTool Use calculator to generate random Use calculator to generate random

numbers from numbers from 1-6. 1-6. 66Ran#+1Ran#+1

TrialTrial One trail is generating 1 random numberOne trail is generating 1 random number Guard:Guard: at cash register: at cash register: Generating 1 and 2Generating 1 and 2at safe: at safe: Generating 3, 4, 5 and 6Generating 3, 4, 5 and 6 Thief: Thief: at cash register: at cash register: Generating 1Generating 1at safe: at safe: Generating 2,3,4,5,6Generating 2,3,4,5,6

Trial no.Trial no. GuardGuard TheifTheif Money Money takentaken

11

22

33

……....

1010

Total Total moneymoney

2 (cash) 3 (safe)

5 (safe) 4 (safe) 0

10,000

3 (safe) 1 (cash) 2000