Probability Theory

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Probability Theory

• Dr. Deshi Ye
• yedeshi_at_zju.edu.cn

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Outline

• 1. Elementary Theorem
• 2. Counting-continuous
• 3. Conditional Probability
• 4. Bayes Theorem

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Element Theorem

• Theorem 3.4.
• If are mutually exclusive
  events in a sample space S, then

Proposition 1.
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Propositions

• Proposition 2.
• If , then
• Proposition 3. If A and B are any events in S,
  then

Proof Sketch 1. Apply the formula of exercise
3.13 (c) and (d) 2. Apply theorem 3.4
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Example

• Suppose that we toss two coins and suppose that
  each of the four points in the sample space
  S(H,H), (H, T), (T, H), (T, T) is equally
  likely and hence has probability ¼. Let
• E (H, H), (H, T) and F (H, H), (T, H).
• What is the probability of P(E or F)?

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Extension
Discussion
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Counting -- continuous

• Binomial theorem
• Multinomial coefficient
• A set of n distinct item is to be divided
  into r distinct groups of respective sizes
  , where
• How many different divisions are possible?

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Multinomial coefficients
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Examples

• In the game of bridge the entire deck of 52 cards
  is dealt out to 4 players. What is the
  probability that
• (a) one of the player receives all 13 spades
• (b) each player receives 1 ace?

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Solution

• (a)

(b)
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EX.

• In the game of bridge the entire deck of 52 cards
  is dealt out to 4 players. What is the
  probability that
• the diamonds ? in 4 players are 6 4 2 1?

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Examples

• A poker hand consists of 5 cards. What is the
  probability that one is dealt a straight?

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Examples

• What is the probability that one is dealt a full
  house?

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Ex.

• If n people are presented in a room, what is
  probability that no two of them celebrate their
  birthday on the same day of the year? How large
  need n be so that this probability is less than
  ½?

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Probability and a paradox

• Suppose we posses an infinite large urn and an
  infinite collection of balls labeled ball number
  1, number 2, and so on.
• Experiment At 1 minute to 12P.M., ball numbered
  1 through 10 are placed in the urn, and ball
  number 10 is withdrawn. At ½ minute to 12 P.M.,
  ball numbered 11 through 20 are placed in the
  urn, and ball number 20 is withdraw. At ¼ minute
  to 12 P.M., and so on.
• Question how many balls are in the urn at 12
  P.M.?

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Paradox
Empty Any number n is withdraw in (1/2)(n-1).
Infinite of course !
Another experiment The balls are withdraw begins
from 1, 2
What is case that arbitrarily choose the withdraw?
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3.6 Conditional probability

• Probability of an event is meaningful iff it
  refers to a given sample.
• P(AS) the probability of A given some space S.
  If respect to more samples.

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Ex.

• 500 machines. Improper assemble I 30Defective
  D 15 Both
• I and D 10.
• P(D)?
• P(DI)?
• P(D and I)?

I 20
D and I 10
D and I 10
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Conditional prob.

• Theorem. If A and B are any events is S and P(B)
  is not empty, the conditional probability of A
  given B is

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Ex.

• A coin is flipped twice. If we assume that all
  four points in the sample space are equally
  likely, what is the probability that both flips
  result in heads, given that the first flip does?

Solution Let E (H,H) be the event that both
flips land heads, and F(H,H), (H,T) denote the
event that the first flip lands heads, then the
desired probability is given by
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Ex.

• Suppose a family has two children. We assume that
  the probability of having a baby boy is ½.
• Now suppose we wish to find the probability that
  the family has one boy and one girl, but we also
  have the information that at least one of the
  children is a boy.
• What is the probability?

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Ex.

• Celine is undecided as to whether to take a
  French course or a chemistry course. She
  estimates that her probability of receiving an A
  grade would be ½ in a French course, and 2/3 in a
  chemistry course.
• If Celine decides to base her decision on the
  flip of a fair coin, what is the probability that
  she gets an A in chemistry?

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Solution

• If we let C be the event that Celine takes
  chemistry and B denote the event that she receive
  an A in whatever course she takes, then the
  desired probability is P(CB).

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Multiplication rule
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Multiplication rule Ex.

• An ordinary deck of 52 playing cards is randomly
  divided into 4 piles of 13 cards each. Compute
  the probability that each pile has exactly 1 Ace?

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Solution another approach

• Define events
• E1 the ace of spades is in any one of the
  piles
• E2 the ace of spades and the ace of hearts are
  in different piles
• E3 the aces of spades, hearts, and diamonds
  are all in different piles
• E4 all aces are in different piles
• P(E1) 1, P(E2E1)39/51, P(E3E1E2)26/50,
• P(E4E1E2E3) 13/49.
• P(E1E2E3E4) P(E1)P(E2E1)P(E3E1E2)P(E4E1E2E3)
  
• 0.105

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Independent

• A is independent of B if and only if
• P(AB)P(A)

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EX.

• Suppose that we toss 2 dice.
• Let E denote the event that the sum of the dice
  is 6 and
• F denote the event that the first die equals 4.
• Q are the events E and F independent?

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Solution

• P(EF) P(4, 2) 1/36.
• P(E) 5/36 P(F) 1/6
• P(E)P(F) 5/216
• Conclusion E and F are not independent!!
• Question how about the event E changed to the
  sum is 7.

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Properties

• If E and F are independent, then so are E and
• If E is independent of F and is also independent
  of G. Is E then necessarily independent of FG?
• Ans No!!

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Generalization

• Theorem 3.8 If A and B are any events in S, then
• if P(A)gt0,P(B)gt0.
• If independent

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Discussion

• P(AB) and P(A n B)
• A, B are independent, mutually exclusive, what is
  the difference?

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Bayes Theorem
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Bayes Theorem

• An experiment depends on the outcomes of various
  intermediate stages.
• EX. An assembly plant receives its voltage
  regulators.
• 60 from B1, 30 from B2, 10 from B3
• Perform according to specifications
• B1 95, B2 80, B3 65.
• Event a voltage regulator received by the plant
  performs according to specifications.

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Tree diagram
P(AB1)0.95
B1
A
0.6
B2
0.3
P(AB2)0. 8
A
0.1
P(AB3)0.65
B3
A
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Solution
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Rule of total probability

• Theorem 3.10.
• If are mutually exclusive events
  of which one must occur, then

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Special case
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EX.

• A laboratory blood test is 95 percent effective
  in detecting a certain disease when it is, in
  fact, present. However, the test also yield a
  false positive result for 1 percent of the
  health person tested.
• Q If .5 percent of the population actually has
  the disease, what is probability a person has the
  disease given that the test result is positive?

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Solution
Only 0.323!

• Let D be the event that the tested person has the
  disease.
• Let E be the event that the test result is
  positive.
• The desired probability is P(DE)

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Bayes Theorem

• Theorem. If are mutually
  exclusive events of which one must occur, then

The prob. Of reaching A via the r-th branch of
the tree
Effect A was caused by the event Br
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EX.
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Summary

• Conditional probability
• Independent of events A, B
• Bayes rule, Bi disjoint

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Homework

• Homework Problems in Textbook (3.76,3.78)  
• Additional Independent trials, consisting of
  rolling a pair of fair dice, are performed. What
  is the probability that an outcome of 5 appears
  before an outcome of 7 when the outcome of a roll
  is the sum of the dice?
• Please using two approaches to solve this
  question. Either independent events or
  conditional probabilities.