Probability

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Probability
2
Probability as a Relative Frequency (occurrence
in the long run)

Tossing a coin
The relative frequency of occurrences of an event
A, should approach the probability P(A), as the
number of trials grows (when the trials are
random and independent of each other).
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The Personal - Probability Interpretation

• Example
• Assume that it is known that the proportion of
  adults who get the flu each winter remains at
  about 30
• Which interpretation of probability applies to
  this statement The probability that I shall get
  the flu this winter is 30?
• Which interpretation of probability applies to
  this statement The probability that a randomly
  selected adult in America will get the flu this
  winter is 30?

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Probability Rules

• Any probability is a number between 0 and 1.
• For any event A,
• Ahead when tossing a coin once
• P(A)0.5 for a fair coin

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Probability Rules

• 1. All possible outcomes together must have
  probability 1

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Rule 1. Probability of a Complement Event

• The probability that an event does not occur is
  1 minus the probability that the event does
  occur.
• Ac is the complement of A the event that A
  does not occur
• P(Ac)1-P(A)
• P(Ac) P(A) 1

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Rule 2. The additive law of probability When A
and B are disjoint or mutually exclusive
B
A
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• 3. Say the probability of getting an A on the
  exam is .25 and the probability of getting a B is
  .30.
• Now find P(A or B)
• Answer
• The events A and B are disjoint (also called
  mutually exclusive), so
• P(A or B) P(A) P(B) .25.30.55

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Rule 3. Independence and the multiplication rule

• Two events are independent if knowing that one
  occurs does not change the probability the other
  occurs.
• If A and B are independent
• This is the multiplication rule for
  independent events.

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Examples

• The probability that a patient is allergic to
  penicillin is 0.20. Suppose this drug is
  administered to 2 patients.
• What is the probability that both patients are
  allergic to penicillin?
• Answer
• Aone of the patients is allergic to penicillin
• Bthe other patient is allergic to penicillin
• P(A)0.2
• P(B)0.2
• In our example the two events are independent,
  since knowing that one patient is allergic
  doesnt change the probability that the other
  patient is allergic.

A and B are independent
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Example

• Let X be the number of boys born in a family of
  2 children. List the possible values of X

BB BG GB GG
2 1 1 0
0.25 0.25 0.25 0.25
The probability of BB is p(first is a boy and
second is a boy) p(first is a boy)
p(second is a boy)0.50.50.25 P(BG)
P(GB) P(GG)
Since the 2 events are independent
0.25
0.25
0.25
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Now we can specify the probability of each value
of X
0.25
0.250.250.5
0.25
Remember that the sum of the probabilities must
equal 1!!!
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Practice

• 1. A probability distribution is given in the
  accompanying table with the additional
  information that the even values of X are equally
  likely. Determine the missing entries of the
  table.

Answer 0.20.20.30.7 The remaining 0.3
probability is equally divided between the values
2,4,and 6
0.1
0.1
0.1
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Probability histogram

• A probability histogram serves as a display of a
  probability distribution
• Example
• Number of boys in a family of 2 children

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Mean of a discrete random variable
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Mean of a discrete random variable

• X x1 x2 x3 xn
• Probability p1 p2 p3 pn
• The mean of a list of numbers is their average
• The mean of a rv is a weighted average of its
  possible values.
• EV x1p1x2p2xnpn
• Mean expected value

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• Example household size in the U.S
• X 1 2 3 4
  5 6 7
• Probability .251 .321 .171 .154 .067
  .022 .014
• Mean of X
• (1)(.251)(2)(.321)(3)(.171) (4)(.154)
  (5)(.067) (6)(.022) (7)(.014)
• 2.587

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Practice mean of rv

• 1. Roll a die once and let X denote the outcome

Calculate the mean outcome of the
die E(X)µX (1)(1/6)(2)(1/6)(3)(1/6)(4)(1/6
)(5)(1/6)(6)(1/6)3.5
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• 2. The California lottery has offered a number of
  games over the years, one such game is Decco, in
  which a player chooses one card from each of the
  four suits in a regular deck of playing cards
  (e.g., 4 of hearts, 3 of clubs, 10 of diamonds
  and jack of spades). A winning card is then drawn
  from each suit. If one or more of the choices
  matches the winning cards drawn, a prize is
  awarded. It costs 1 for each play. The following
  table lists the possible prizes in this game,
  taken from the back of a deco card

What is the probability that the player will win
nothing? P (Number of matches 0)
1-(.000035.00168.0303.242).726
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• 2. The California lottery has offered a number of
  games over the years, one such game is Decco, in
  which a player chooses one card from each of the
  four suits in a regular deck of playing cards
  (e.g., 4 of hearts, 3 of clubs, 10 of diamonds
  and jack of spades). A winning card is then drawn
  from each suit. If one or more of the choices
  matches the winning cards drawn, a prize is
  awarded. It costs 1 for each play. The following
  table lists the possible prizes in this game,
  taken from the back of a deco card

What is the expected value of X (how much would
you win or lose per ticket in this game, over the
long run)? E(X)
4,999(.000035)49(.00168)4(.0303)0(.242)(-1)(.7
26)-.3475
Question How much should a game cost for it to
be a fair game?
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• 3. The New Jersey lottery the players picks six
  numbers from the choices 1 to 49. Six winning
  numbers are selected. If the player matches at
  least 3 out of the 6 winning numbers the ticket
  is a winner (wins 3) matching 4 or more results
  in a prize determined by the number of successful
  entries. The probability of winning anything is
  1/54. How many times would you play before
  winning anything (and with what probability)?

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Practice

• The probability that a selected Caucasian
  American child will have blond or red hair is
  23. The probability of having blond hair is 14.
• a) What is the probability of having red hair?
• b) What is the probability of having neither
  blond nor red hair?
• 2. What is wrong with each of the following
  statements
• a) The probability a randomly selected driver
  will wear seat belt is .75, whereas the
  probability that s/he will not be wearing one is
  .30.
• b) The probability that a randomly selected car
  is red is 1.20.

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Practice - Continued

• 3. A study by Kahneman and Tversky (1982) asked
  people Linda is 31 years old, single, outspoken
  and very bright. She majored in philosophy. As a
  student, she was deeply concerned with issues of
  discrimination and social justice, and also
  participated in antinuclear demonstrations
  Please check off the most likely alternative
• a) Linda is a bank teller
• b) Linda is a bank teller and is active in the
  feminist movement

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Practice Continued -2

• 3. A study by Kahneman and Tversky (1982) asked
  people Linda is 31 years old, single, outspoken
  and very bright. She majored in philosophy. As a
  student, she was deeply concerned with issues of
  discrimination and social justice, and also
  participated in antinuclear demonstrations
  Please check off the most likely alternative
• a) Linda is a bank teller
• b) Linda is a bank teller and is active in the
  feminist movement
• 2. What is wrong with each of the following
  statements
• a) The probability a randomly selected car is
  red is .20 and the probability a randomly
  selected car is a red sport car is .25

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Rule 4. The subset rule

• If the ways in which an event B can occur are a
  subset of those for event A then