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Title: Probabilities and Proportions

1
Probabilities and Proportions
2
Lotto

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?

3
Roulette

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.

4
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 ?

5
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.

6
TV Game Show

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?

7
Lets 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/LetsMakeaDea
l.html

8
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
9
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

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.
Result A B C
Lose Car Goat Goat
Win Goat Car Goat
Win Goat Goat Car
Monty will show door B, you switch to C and win.
P(WIN) 2/3 !!!!
10
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!

11
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

12
Probability

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?
?????

13
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

14
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)

15
Probabilities and Proportions

Probabilities and proportions
are 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 is
left handed with a probability
of approximately 0.1.

16
Where do probabilities come from?

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)

17
Where do probabilities come from?

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.

18
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)

19
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

20
Simple Probability Models

The probability that an event A occurs
is written in shorthand as P(A).

21
Example Sum of two die
22
Example Sum of two die
23
Example Sum of two die
24
Example Sum of two die
25
House Sales Example

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

26
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

27
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

28
House Sales Example

For a sale selected at random from these 600
sales, find the probability that the sale was
over 400,000, i.e. event A occurs.

P(A) 61/600 0.102
29
House Sales Example

For a sale selected at random from these 600
sales, find the probability that the sale was
not over 400,000, i.e. occurs.

P( ) (155384)/600 539/600 0.898 Note
that P(A) P( ) 1 and that P( ) 1 P(A)
30
House Sales Example

For 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
31
House Sales Example

For a sale selected at random from these 600
sales, find the probability that the sale was
d) made within 45 days and sold for over
400,000, i.e. both B and A occur.

P(B and A) 20/600 0.033
32
House Sales Example

For a sale selected at random from these 600
sales, find the probability that the sale was
e) made within 45 days and/or sold for over
400,000, i.e. either A or B occur.

33
House Sales Example

For a sale selected at random from these 600
sales, find the probability that the sale was
e) made within 45 days and/or sold for over
400,000.

P(B and/or A) (246 61 20)/600 287/600
0.478
34
House Sales Example

For a sale selected at random from these 600
sales, find the probability that the sale was
f ) on the market for less than 45 days given
that it sold for over 400,000

35
House Sales Example

For a sale selected at random from these 600
sales, find the probability that the sale was
f ) on the market for less than 45 days given
that it sold for over 400,000

P(B given A) P(BA) 20/61 0.328
36
Conditional Probability