C4, L1, S1 Probabilities and Proportions. C4, L1, S2 I am offered two lotto cards: –Card 1: has numbers –Card 2: has numbers Which card should I take

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Probabilities and Proportions

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I am offered two lotto cards:

– Card 1: has numbers

– Card 2: has numbers

Which card should I take so that I have the greatest chance of winning lotto?

Lotto

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In the casino I wait at the roulette wheel until I see a run of at least five reds in a row.

I then bet heavily on a black.

I am now more likely to win.

Roulette

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Coin Tossing

I am about to toss a coin 20 times.

What do you expect to happen?

Suppose that the first four tosses have been heads and there are no tails so far. What do you expect will have happened by the end of the 20 tosses ?

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Coin Tossing

• Option A– Still expect to get 10 heads and 10 tails. Since

there are already 4 heads, now expect to get 6 heads from the remaining 16 tosses. In the next few tosses, expect to get more tails than heads.

• Option B– There are 16 tosses to go. For these 16 tosses I

expect 8 heads and 8 tails. Now expect to get 12 heads and 8 tails for the 20 throws.

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• In a TV game show, a car will be given away.

– 3 keys are put on the table, with only one of them

being the right key. The 3 finalists are given a

chance to choose one key and the one who

chooses the right key will take the car.

– If you were one of the finalists, would you prefer

to be the 1st, 2nd or last to choose a key?

TV Game Show

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Let’s Make a Deal Game Show

• You pick one of three doors – two have booby prizes behind them– one has lots of money behind it

• The game show host then shows you a booby prize behind one of the other doors

• Then he asks you “Do you want to change doors?”– Should you??! (Does it matter??!)

• See the following website:• http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html

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Game Show Dilemma

Suppose you choose door A. In which case

Monty Hall will show you either door B or C

depending upon what is behind each.

No Switch Strategy ~ here is what happens

Result A B C

Win Car Goat Goat

Lose Goat Car Goat

Lose Goat Goat Car

P(WIN) = 1/3

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Game Show Dilemma

Suppose you choose door A, but ultimately

switch. Again Monty Hall will show you either

door B or C depending upon what is behind each.

Switch Strategy ~ here is what happens

Result A B C

Lose Car Goat Goat

Win Goat Car Goat

Win Goat Goat Car

Monty will show either B or C.

You switch to the one not shown

and lose.

Monty will show door C, you switch to B and win.

Monty will show door B, you switch to C and win.

P(WIN) = 2/3 !!!!

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Matching Birthdays• In a room with 23 people what is the

probability that at least two of them will have the same birthday?

• Answer: .5073 or 50.73% chance!!!!!

• How about 30? • .7063 or 71% chance!• How about 40? • .8912 or 89% chance!• How about 50? • .9704 or 97% chance!

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Probability

In this section we will…– Introduce us to basic ideas about probabilities:

• what they are and where they come from• simple probability models• conditional probabilities• independent events

– Teach us how to calculate probabilities:• through tables of counts and probability tables

for independent events

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I toss a fair coin (where fair means ‘equally likely outcomes’)

• What are the possible outcomes?

Head and tail• What is the probability it will turn up heads?

1/2

I choose a person at random and check which eye she/he winks with• What are the possible outcomes?

Left and right• What is the probability they wink with their left eye?

?????

Probability

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What are Probabilities?

• A probability is a number between 0 & 1 that quantifies uncertainty

• A probability of 0 identifies impossibility

• A probability of 1 identifies certainty

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Where do probabilities come from?

• Probabilities from models:The probability of getting a four when a fair dice is rolled is

1/6 (0.1667 or 16.7%)

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Probabilities and ProportionsProbabilities and proportionsare numerically equivalent.(i.e. they convey the same information.)

e.g. The proportion of U.S. citizens who are left handed is 0.1; a randomly selected U.S. citizen isleft handed with a probabilityof approximately 0.1.

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• Probabilities from data– In a survey conducted by students in a STAT 110

course there were 348 WSU students sampled. – 212 of these students stated they regularly drink

alcohol. – The estimated probability that a randomly chosen

Winona State students drinks alcohol is

212/348 (0.609, 60.9%)


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• Subjective Probabilities– The probability that there will be another

outbreak of ebola in Africa within the next year is 0.1.

– The probability of rain in the next 24 hours is very high. Perhaps the weather forecaster might say a there is a 70% chance of rain.

– A doctor may state your chance of successful treatment.


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Simple Probability Models

Terminology:

• a random experiment is an experiment whose outcome cannot be predicted– E.g. Draw a card from a well-shuffled

pack

• a sample space is the collection of all possible outcomes– 52 outcomes (AH, 2H, 3H, …, KH,…,

AS, …,KS)

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Simple Probability Models• an event is a collection of outcomes

– E.g. A = card drawn is a heart

• an event occurs if any outcome making up that event occurs– drawing a 5 of hearts

• the complement of an event A is denoted as , it contains all outcomes not in A Eg = card drawn is not a heart

= card drawn is a spade, club or

diamond

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For equally likely outcomes, and a given event A:

Simple Probability Models

“The probability that an event A occurs”

is written in shorthand as P(A).

P(A) =Number of outcomes in A

Total number of outcomes

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Example: Sum of two die

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1. House Sales Example

Below is a table containing some information for a sample of 600 sales of single family houses in 1999.

Days on the Market

Less than 45 Days

45 - 90 Days More than 90 Days

Under $250,000 75 77 3 155

$250,000 - 400,000 151 192 41 384

Over $400,000 20 31 10 61

246 300 54 600

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1. House Sales Example• Let A be the event that a sale is

over $400,000

– is the event that a sale is NOT over $400,000

Days on the Market

Less than 45 Days


Under $250,000 75 77 3 155

$250,000 - 400,000 151 192 41 384

Over $400,000 20 31 10 61

246 300 54 600

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1. House Sales Example• B be the event that a sale is

made within 45 days– So is the event that a sale takes

longer than 45 days

Days on the Market

Less than 45 Days


Under $250,000 75 77 3 155

$250,000 - 400,000 151 192 41 384

Over $400,000 20 31 10 61

246 300 54 600

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For a sale selected at random from these 600 sales, find the probability that the sale was:

a) over $400,000, i.e. event A occurs.

P(A) = 61/600 = 0.102

Days on the Market

Less than 45 Days


Under $250,000 75 77 3 155

$250,000 - 400,000 151 192 41 384

Over $400,000 20 31 10 61

246 300 54 600

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1. House Sales ExampleFor a sale selected at random from these 600 sales, find the probability that the sale was:

b) not over $400,000, i.e. occurs.

P( ) = (155+384)/600 = 539/600 = 0.898* Note that P(A) + P( ) = 1 and that P( ) = 1 – P(A)Days on the Market

Less than 45 Days


Under $250,000 75 77 3 155

$250,000 - 400,000 151 192 41 384

Over $400,000 20 31 10 61

246 300 54 600

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1. House Sales ExampleFor a sale selected at random from these 600 sales, find the probability that the sale was:

c) made in 45 days or more, i.e. occurs.

P( ) = (300 + 54)/600 = 354/600 = 0.59

Days on the Market

Less than 45 Days


Under $250,000 75 77 3 155

$250,000 - 400,000 151 192 41 384

Over $400,000 20 31 10 61

246 300 54 600

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d) made within 45 days and sold for over $400,000, i.e. both B and A occur.

P(B and A) = 20/600 = 0.033Days on the Market

Less than 45 Days


Under $250,000 75 77 3 155

$250,000 - 400,000 151 192 41 384

Over $400,000 20 31 10 61

246 300 54 600

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e) made within 45 days and/or sold for over $400,000, i.e. either A or B occur.

Days on the Market

Less than 45 Days


Under $250,000 75 77 3 155

$250,000 - 400,000 151 192 41 384

Over $400,000 20 31 10 61

246 300 54 600

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Days on the Market

Less than 45 Days


Under $250,000 75 77 3 155

$250,000 - 400,000 151 192 41 384

Over $400,000 20 31 10 61

246 300 54 600

P(B and/or A) = (246 + 61 – 20)/600 = 287/600 = 0.478


e) made within 45 days and/or sold for over $400,000.

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f ) on the market for less than 45 days given that it sold for over $400,000

Days on the Market

Less than 45 Days


Under $250,000 75 77 3 155

$250,000 - 400,000 151 192 41 384

Over $400,000 20 31 10 61

246 300 54 600

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f ) on the market for less than 45 days given that it sold for over $400,000

Days on the Market

Less than 45 Days


Under $250,000 75 77 3 155

$250,000 - 400,000 151 192 41 384

Over $400,000 20 31 10 61

246 300 54 600

P(B given A) = P(B|A) = 20/61 = 0.328

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Conditional Probability

• The sample space is reduced.

• Key words that indicate conditional probability are:“given that”, “of those”, “if …”,

“assuming that”

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“The probability of event A occurring given that event B has already occurred”

is written in shorthand as P(A|B)

It is defined as follows:

P(A|B) = P(A and B)/P(B)

Conditional Probability

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1. House Sales ExampleFor a sale selected at random from these 600 sales,

g) What proportion of the houses that sold in less than 45 days, sold for more than $400,000?

Days on the Market

Less than 45 Days


Under $250,000 75 77 3 155

$250,000 - 400,000 151 192 41 384

Over $400,000 20 31 10 61

246 300 54 600

C4, L1, S39

1. House Sales ExampleFor a sale selected at random from these 600 sales,

g) What proportion of the houses that sold in less than 45 days, sold for more than $400,000?

Days on the Market

Less than 45 Days


Under $250,000 75 77 3 155

$250,000 - 400,000 151 192 41 384

Over $400,000 20 31 10 61

246 300 54 600

P (A|B) = 20/246 = 0.081

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1. Heart Disease

In 1996, 6631 New Zealanders died from coronary heart disease. The numbers of deaths classified by age and gender are:

Sex

Age Male Female Total

< 45 79 13 92

45 - 64 772 216 988

65 - 74 1081 499 1580

> 74 1795 2176 3971

Total 3727 2904 6631

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Let

A be the event of being under 45B be the event of being maleC be the event of being over 64

1. Heart Disease

Sex


< 45 79 13 92

45 - 64 772 216 988

65 - 74 1081 499 1580

> 74 1795 2176 3971

Total 3727 2904 6631

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Find the probability that a randomly chosen member of this population at the time of death was:

a) under 45 P(A) = 92/6631 = 0.0139

1. Heart Disease

Sex


< 45 79 13 92

45 - 64 772 216 988

65 - 74 1081 499 1580

> 74 1795 2176 3971

Total 3727 2904 6631

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1. Heart Disease

Sex


< 45 79 13 92

45 - 64 772 216 988

65 - 74 1081 499 1580

> 74 1795 2176 3971

Total 3727 2904 6631


b) male assuming that the person was younger than 45.

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Sex


< 45 79 13 92

45 - 64 772 216 988

65 - 74 1081 499 1580

> 74 1795 2176 3971

Total 3727 2904 6631


b) male given that the person was younger than 45. P(B|A) = 79/92 = 0.8587

2. Heart Disease

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Sex


< 45 79 13 92

45 - 64 772 216 988

65 - 74 1081 499 1580

> 74 1795 2176 3971

Total 3727 2904 6631


c) male and was over 64.P(B and C) = (1081 + 1795)/6631= 2876/6631

1. Heart Disease

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Sex


< 45 79 13 92

45 - 64 772 216 988

65 - 74 1081 499 1580

> 74 1795 2176 3971

Total 3727 2904 6631


d) over 64 given they were female.

1. Heart Disease

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Sex


< 45 79 13 92

45 - 64 772 216 988

65 - 74 1081 499 1580

> 74 1795 2176 3971

Total 3727 2904 6631

P(C|B) = (499+2176)/2904 = 0.9211

1. Heart DiseaseFind the probability that a randomly chosen member of this population at the time of death was:

d) over 64 given they were female.

Documents

C4, L1, S1 Probabilities and Proportions. C4, L1, S2 I am offered two lotto cards: –Card 1: has numbers –Card 2: has numbers Which card should I take