1. Probabilities and Proportions

2. Lotto I am offered two lotto cards: Card 1: 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 Option B
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. 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:

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
Lose
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
Win
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: 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 What is Chapter 3 trying to do?
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
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 ( or 16.7%)

15. 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%)

16. 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.

17. 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)

18. 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

19. Simple Probability Models
“The probability that an event A occurs”
is written in shorthand as P(A).
For equally likely outcomes, and a given event A:
P(A) =
Number of outcomes in A
Total number of outcomes

20. Conditional Probability
The sample space is reduced.
Key words that indicate conditional probability are:
“given that”, “of those”, “if …”, “assuming that”

21. Conditional Probability
“The probability of event A occurring given that event B has already occurred”
is written in shorthand as P(A|B)

22. 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
772
216
988
1081
499
1580
> 74
1795
2176
3971
3727
2904
6631

23. 1. Heart Disease Let A be the event of being under 45
B be the event of being male
C be the event of being over 64
Sex
Age
Male
Female
Total
< 45 79 13 92
772
216
988
1081
499
1580
> 74
1795
2176
3971
3727
2904
6631

24. 1. Heart Disease
Find the probability that a randomly chosen member of this population at the time of death was:
under 45
P(A) = 92/6631 =
Sex
Age
Male
Female
Total
< 45 79 13 92
772
216
988
1081
499
1580
> 74
1795
2176
3971
3727
2904
6631

25. 1. Heart Disease
Find the probability that a randomly chosen member of this population at the time of death was:
male assuming that the person was younger than 45.
Sex
Age
Male
Female
Total
< 45 79 13 92
772
216
988
1081
499
1580
> 74
1795
2176
3971
3727
2904
6631

26. Heart Disease
Find the probability that a randomly chosen member of this population at the time of death was:
male given that the person was younger than 45.
P(B|A) = 79/92 =
Sex
Age
Male
Female
Total
< 45 79 13 92
772
216
988
1081
499
1580
> 74
1795
2176
3971
3727
2904
6631

27. 1. Heart Disease
Find the probability that a randomly chosen member of this population at the time of death was:
c) male and was over 64.
P(B and C) = ( )/6631= 2876/6631
Sex
Age
Male
Female
Total
< 45 79 13 92
772
216
988
1081
499
1580
> 74
1795
2176
3971
3727
2904
6631

28. 1. Heart Disease
Find the probability that a randomly chosen member of this population at the time of death was:
d) over 64 given they were female.
Sex
Age
Male
Female
Total
< 45 79 13 92
772
216
988
1081
499
1580
> 74
1795
2176
3971
3727
2904
6631

29. 1. Heart Disease
Find the probability that a randomly chosen member of this population at the time of death was:
d) over 64 given they were female.
P(C |B) = ( )/2904 =
Sex
Age
Male
Female
Total
< 45 79 13 92
772
216
988
1081
499
1580
> 74
1795
2176
3971
3727
2904
6631