Part 1 Definitions

1
Probability

• Part 1 Definitions
• Event
• Probability
• Union
• Intersection
• Complement
• Part 2 Rules

2
Definitions - Event

• An event is a specific collection of possible
  outcomes.
• For example, the event A could be the event
  that one Head occurs in two tosses of a coin.

HH HT TH TT
A
3
Definitions - Probability

• The probability of an event is the ratio of the
  number of outcomes matching the event description
  to the number of possible outcomes.
• P(1 head in 2 tosses) 2/4 .5
• Probabilities are ALWAYS between 0 and 1.

4
Definitions - Union

• The union of two events A and B is the event that
  occurs if either A or B or both occur on a single
  measurement.
• For example, suppose
• A a car has two doors
• B a car is red
• Then, if we randomly select a car from a large
  parking lot, P(A U B) is the probability that
  the car is either a two-door model or red or both
  two-door and red.

5
Definitions Union

• Define
• C being female
• D being nineteen years old
• What is P(C U D) for a randomly selected person
  from among the set of people currently in this
  room?

6
Definitions Union

• of women ________
• plus
• of 19 year olds ________
• minus
• of 19 year old women ________
• equals
• either 19 or a woman or both ________

7
Definition Union

• Total of people in this room _________
• Therefore, P(C U D) __________

8
Definitions Intersection

• The intersection of two events A and B is the
  event that occurs if and only if both A and B
  occur on a single measurement.
• Suppose we randomly select a person from the set
  of people currently in this room. With A and B
  defined as before, the probability that a
  randomly-selected person is a 19 year old woman
  is
• P(A n B)

9
Definitions Complement

• The complement of an event A is the event that A
  does not occur.

A
HH TT
HT TH
A
10
Definitions Complement

• P(A) 1 P(A)
• P(A) P(A) 1
• The second equation says, Either A happens or A
  doesnt happen. There are no other
  possibilities. That may seem obvious, but keep
  it in mind on exams.

11
Part 2 Rules

• Additive Rule
• Mutually Exclusive Events
• Conditional Probability
• Multiplicative Rule
• Independence

12
Rules Additive Rule

• P(A U B) P(A) P(B) P(A n B)
• Recall 19 year old women example.
• When you add of 19 year olds to of women,
  you count the 19 year old women in both groups
  so you count them twice. Subtract that number
  once as a correction.

13
Rules Mutually Exclusive Events

• Two events are mutually exclusive when
• P(A n B) 0
• and
• P(A U B) P(A) P(B)
• Be careful! Note that the first equation uses n
  while the second one uses U.

14
Rules Conditional Probability

• When you have information that reduces the set of
  possible outcomes, you work with new
  probabilities that are conditional on that new
  information.
• Remember that the of possible outcomes is the
  denominator of the ratio that gives probability
  of an event.
• The probability changes on the basis of new
  information because the numerator stays the same
  but the denominator decreases.

15
Rules Conditional Probability

• Example Suppose before class I picked a card at
  random from a standard deck of cards. What is
  P(E) if
• E Card I picked is a Club?
• Note that this is a question about the ordinary
  (non-conditional) probability.

16
Rules Conditional Probability

• P(E) 13/52 .25 (Do you see why?)
• Now, suppose I tell you that the card I picked is
  black. What is the conditional probability that
  the card is a club given that it is black?
• P(E black) 13/26 .5
• Only 26 cards in a normal deck are black.

17
Rules Conditional Probability

• We write
• P(A B) P(A n B)
• P(B)

18
A
B
A n B
P(A B) P(A n B) P(B)
19
Rules Multiplicative Probability

• P(A n B) P(A) P(BA)
• P(A n B) P(B) P(AB)

20
Rules Multiplicative Probability

• To see why, begin with the conditional
  probability formula, and multiply both sides by
  either P(A) or P(B)
• P(A B) P(A n B)
• P(B)
• P(B A) P(A n B)
• P(A)

21
Rules - Independence

• Events A and B are independent if the occurrence
  of one does not alter the probability of the
  other.
• P(AB) P(A)
• P(BA) P(B)

22
Rules Independence

• We can now re-write the multiplicative rule for
  the special case of independent events
• P(A n B) P(A) P(B)
• This is because P(BA) P(B) for independent
  events.

23
Probability Examples

• 60 of Western students are female. 60 of female
  students have a B average or better. 80 of male
  students have less than a B average.
• a. What is the probability that a randomly
  selected student will have less than a B average?

24
A Being female B Having a B average or better
B
.6 x .6 .36 .6 x .4 .24 .4 x .2
.08 .4 x .8 .32
.6
A
.6
.4
B
B
.2
.4
A
B
.8
25
Probability Examples

• a. What is the probability that a randomly
  selected student will have less than a B average?
• P(B) P(B n A) P(B n A)
• Either we get (B and A) or we get (B and A).
• That is, either our randomly selected student
  who has less than a B average is a female or he
  is a male.

26
Probability Examples

• P(B) P(B n A) P(B n A)
• By Multiplicative Rule
• P(B) P(BA)P(A) P(BA)P(A)
• (.4.6) (.8.4)
• .56

27
Reminder Multiplicative Probability

• P(A n B) P(A) P(BA)
• P(A n B) P(B) P(AB)

28
Probability Examples

• b. If we randomly select a student at Western and
  note that this student has a B or better average,
  what is the probability that the student is male?

29
A Being female B Having a B average or better
B
.6 x .6 .36 .6 x .4 .24 .4 x .2
.08 .4 x .8 .32
.6
A
.6
.4
B
B
.2
.4
A
B
.8
30
Probability Examples

• We know that overall, 40 of students (A) are
  male. But among the students who get a B or
  better average, what proportion are male?
• The probability of getting a B or better average
  is .44 (from .36 for women and .08 for men).
  Thus,
• P(AB) P(A n B) .08 .1818
• P(B) .44

31
Probability Examples

• Youre on a game show. Youre given a choice of 3
  doors you can open. Behind one door is a car.
  Behind each of the other two doors is a goat. You
  win what is behind the door you open. You pick a
  door, but dont get to open it yet. The host
  opens another door, behind which is a goat. The
  host then says to you, Do you want to stick with
  the door you chose or switch to the other
  remaining door?
• Is it to your advantage to switch?

32
Probability Examples

• Yes you should switch to the other door. The
  door you originally chose has a 1/3rd chance of
  winning the car. The other remaining door has a
  2/3rd chance of winning the car.

33
Probability Examples

• Suppose you originally pick Door 1. The door the
  host opens is indicated by the boldface type
  below. What you win if you switch is underlined.
• Door 1 Door 2 Door 3
• Goat Goat Car
• Goat Car Goat
• Car Goat Goat

34
Probability - Examples

• When you picked Door 1, there was a 2/3rd
  probability that the car was behind one of the
  other doors.
• That is still true after the host opens one of
  those other doors.
• But since the host knows where the car is, he
  opens a door that has a goat behind it. So now
  the 2/3rd probability is all associated with the
  one remaining door.
• The critical point is that the host has
  information so we are dealing with the
  conditional probability of the car being behind
  Door 3 GIVEN THAT the host opened Door 2 (after
  you picked Door 1).