Probability

1
Probability

• Likelihood (chance) that an event occurs
• Classical interpretation of probability all
  outcomes in the sample space are equally likely
  to occur (random sampling)
• Empirical probability conduct actual experiments
  to get the likelihood
• Subjective probability ask professors, friends,
  mom, etc.

2
Basic Concepts

• Probability experiment a chance process that
  leads to well-defined results (outcomes)
• Outcome the distinct possible result of a single
  trial of a probability experiment
• Sample space the set of possible outcomes
• Event identified with certain of outcomes

3
Sample space

• Example 4-2 on page 180
• Sample space 52 outcomes
• Event Queen 4
• Event Heart 12
• Event King Spade 1
• Example 4-4
• Sample space 8
• Event Exactly two boys 3

4
Tossing a Coin

• Tossing a coin once head (H) or tail (T)
• Tossing two times HH, HT, TH, TT
• Tossing three times HHH, HHT, HTH, HTT, THH,
  THT, TTH, TTT? 2 X 2 X 2

5
Tree Diagram

• H head, T tail

6
Rolling a Die

• Rolling once 1, 2, 3, 4, 5, 6
• Rolling twice (1, 1), (1,2) (2, 1), (2, 2),
  (6,6)?62
• Rolling three times (1,1,1), (1,1,2) (1,2,1)
  (1,6,6), (2,1,1)(2,1,2)(2,6,6),
  (3,1,1)(6,6,6,)?63
• Rolling four times How to get the sample space?

7
Combination

• Selecting r distinct objects out of n objects
  regardless of order at a time
• Example select two students for awards among 5
  students
• N factorial n! n X (n-1) X (n-2) X 1
• 0! 1

8
Permutation

• An arrangement of n objects in a specific order
  using r objects at a time.
• Taking r ordered objects out of n objects at a
  time.
• Selecting one student for 10K award and another
  for 5K award among 5 students.

9
Classical Probability

• P(E) is the probability that the event E occurs
  expected (not actualized) likelihood
• The number of outcomes of event E, NE, divided by
  the number of total outcomes in the sample space,
  N.

10
Probability Rules

• P(E) is a number between 0 and 1
• Probability zero, P(E)0, means the event will
  not occur.
• Probability 1, P(E)1, means only the event
  occurs all the times.
• Sum of the probabilities of all outcomes in the
  sample space is 1

11
Complementary Events

• the set of outcomes in the sample space that are
  not included in the outcome of E
• P(E) 1 - P(E)
• P(E) 1 - P(E)
• P(E) P(E) 1

12
Empirical Probability

• Is your quarter really fair? Hmm I guess the
  probability of head is larger than ½ for some
  reason.
• How about your die? Do all 1 through 6 have the
  equal chance of 1/6 to be selected?
• How can you check that?

13
Addition Rule

• Probability that event A or B occurs
• P(A or B) P(A) P(B) P (A and B)
• P(A U B) P(A) P(B) P (A n B)
• P(Nurse or Male)P(N)P(M)-P(N and M), Figure
  4-5, p.198.
• Question 15, p.200.

14
Mutually Exclusive Events

• P(A U B) P(A) P(B) - 0
• P (A n B) 0
• P(Monday or Sunday)P(Monday)P(Sunday)-0

15
Multiplication Rule

• Probability that both events A and B occur
• P(A n B) P(A) X P(B)
• Example 4-24, p.206
• P(queen and ace) P(queen) X P(ace) 4/52 X
  4/52
• Example 4-25
• P(blue and white)P(blue) X P(white) 2/10 X
  5/10
• What if event A and B are related?

16
Statistical Independence

• Occurrence of an event does not change the
  probability that other events occur.
• Occurrence of one measurement in a variable
  should be independent of the occurrence of
  others.
• Drawing a card with/without replacement.
• With replacement-gtindependent (Ex. 4-25)
• Without replacement-gtdependent
• How do we know if two events are statistically
  independent?

17
Examples

• How to put an elephant into a refrigerator?
• Open the door
• Put an elephant into the refrigerator
• Close the door
• Now, how to put an hippo into the refrigerator?
• What makes a difference?
• Question 1, p.215

18
Statistical Dependence 1

• Example 4-25, pp.206-207
• What if no replacement?
• Suppose a blue ball is selected in the 1st trial
• P(blue) is 2/10 in the 1st trial

1st Trial 2nd Trial
P(Blue) 2/10 1/9
P(Red) 3/10 3/9
P(White) 5/10 5/9
19
Statistical Dependence 2

• P(blue then white)2/10 X 5/10 w/o replacement
• P(blue then white)2/10 X 5/9 w/ replacement
• 5/9 probability that event B (white ball) occurs
  given event A (blue) already occurred.
• Figure 4-6. p. 210.

1st Trial 2nd Trial
P(Blue) 2/10 1/9
P(Red) 3/10 3/9
P(White) 5/10 5/9
20
Conditional Probability

• P(BA) is the probability that event B occurs
  after event A has already occurred.
• P(BA)P(A n B) / P(A)
• P(A n B) P(A) X P(BA) in case of statistical
  dependence

21
Statistical Independence, again

• Events A and B are statistically independent, if
  and only If P(BA)P(B) or P(AB)P(A)
• Example 4-34, p.211
• P(YesFemale)P(Female n Yes) / P(Female)
  8/100/50/100 8/50 ? 40/100
• Events Female and Yes are not independent
• P(A n B) P(A) X P(BA)50/1008/508/100
• P(A n B) P(A) X P(B) in case of statistical
  independence because P(BA)P(B)

Yes No Total
Male 32 18 50
Female 8 42 50
40 60 100
22
Examples Example 4-25, p207

• With Replacement
• P(WB)P(W n B)/P(B) 2/105/10/2/105/10P(W)
  
• Events white (2nd trial) and blue (1st trial) are
  independent
• Without Replacement
• P(WB)P(W n B)/P(B)2/105/9/2/10 5/10 ?
  P(W)
• Events white (2nd trial) and blue (1st trial) are
  dependent
• Event blue in the 1st trial influences the
  probability of event white in the 2nd trial.

23
Examples Question 34, p216

• P(opposefreshman)27/80/50/8027/50
• P(sophomorefavor)23/80/38/8023/38
• P(No opinionsophomore)?
• P(Favor freshman)?

Favor Oppose No opinion Total
Freshman 15 27 8 50
Sophomore 23 5 2 30
Total 38 32 10 80
24
Summary

• Addition probability that event A or B occurs
• P(A U B) P(A) P(B) P (A n B)
• P (A n B) 0 if mutually exclusive
• Multiplication probability that both events A
  and B occur
• P(A n B) P(A) X P(BA)
• P(BA)P(B) if statistically independent