Week 4

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Week 4
Probability The Study of Randomness
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OBJECTIVES

• To define the term probability.
• To discuss the classical, the relative frequency,
  and the subjective approaches to probability.
• To understand the terms experiment, event, and
  outcome.
• To define the terms random variable and
  probability distribution
• To distinguish between a discrete and continuous
  probability distribution

• To calculate the mean, variance, and standard
  deviation of a discrete probability distribution.

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Why study probability?

• Probability plays a central role in inferential
  statistics because there is a relationships
  between samples and populations.
• Probability allows us to start with a population
  and predict what kind of sample is likely to be
  obtained from it.
• To obtain probabilistic statements about
  populations parameters (Estimation Hypothesis
  Testing)

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Why study probability?

• If we know the blood types of a man and a woman,
  what can we say about the blood types of their
  future children?
• Give a test for the AIDS virus to the employees
  of a small company. What is the chance of at
  least one positive test if all the people tested
  are free of the virus?
• A man experiences an acute myocardial infarction
  while dining with his family.
• What is the chance that he will die within
  the first hour after the infarction?

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1. Experiment, Event and Sample Space

• Experiment A process of observation with an
  uncertain outcome.
• Event Experimental outcome that may or may not
  occur.
• Sample Space The set of all possible outcomes.

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Examples

• Experiment 1 Toss a coin twice.
• Sample space S - HH, HT, TH, TT.
• Event - At least one head HH, HT,
  TH.

• Experiment 2 Spin a roulette wheel.
• Sample space S 1, 2. , 36
• Event - The roulette wheels stops at even
  number

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2. PROBABILITY

• Number between 0 and 1, inclusive, that indicates
  how likely event is to occur
• (What do you think about this definition?)
• The closer a probability is to 0, the more
  improbable it is that the associated event will
  occur.
• The closer the probability is to 1, the more sure
  we are that the associated event will happen
• Null events
• Certain events

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3. APPROACHES TO PROBABILITY

• Classical probability
• Subjective probability
• Relative Frequency Concept

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APPROACHES TO PROBABILITY

• 3.1 Classical probability is based on the
  assumption that the outcomes of an experiment are
  equally likely.
• Using this classical viewpoint, then
• 
  

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EXAMPLE 1

• Consider the experiment of tossing two coins
  once.
• The sample space S HH, HT, TH, TT
• Consider the event of one head.
• Probability of one head 2/4 1/2.

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3.2 SUBJECTIVE PROBABILITY

• 3.2 Subjective probability The likelihood
  (probability) of a particular event happening
  that is assigned by an individual based on
  whatever information is available.
• Examples of subjective probability are
• A physician may say that s/he is 95 certain that
  a patient has a particular disease.
• For a physician the probability that a patient
  will be cured when treated with a certain
  antibiotic is 0.75.

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3.3 RELATIVE FREQUENCY CONCEPT

• Relative frequency is another term for
  proportion.
• It can be computed from the formula

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RELATIVE FREQUENCY CONCEPT

• Buffon, the French naturalist, tossed a coin 4040
  times.
• Results 2048 heads gt Relative frequency
  2048/4040 0.5069

• Karl Pearson, around 1900, tossed a coin 24,000
  times.
• Results 12,012 heads gt Relative frequency
  0.5005

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RELATIVE FREQUENCY CONCEPT

• John Kerrich, an Australian mathematician, while
  imprisoned during World War II tossed a coin
  10,000 times.
• Results 5067 heads gt Relative frequency
  0.5067.

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RELATIVE FREQUENCY CONCEPT

• If an experiment is repeated many, many times
  without changing the experimental conditions, the
  relative frequency of any particular event will
  settle down to some value.
• The probability can be defined as the limiting
  value of the relative frequency when number of
  trials approaches infinity.

Probability is a long-term relative frequency.
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EXAMPLE 2

• The question was whether O. J. Simpson was
  guilty. In a sample of 500 students on the Levels
  campus, 275 indicated that the defendant was
  guilty. What is the probability that a student
  on this campus will indicate that the defendant
  in this case was guilty?
• P(guilty) 275/500 0.55.

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4. RecapSome properties of probabilities

• Probabilities have the following properties
• 1. P(A) is always a value between 0 and 1.
• 2. P(A) 0 means A never occurs
• 3. P(A) 1 means A always occurs
• The collection of all possible outcomes has
  probability 1 gt P(S) 1

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Lecture Exercise 1

• A roulette wheel has 38 slots, numbered 0,
  00, and 1 to 36. The slots 0 and 00 are coloured
  green, 18 of the others are red, and 18 are
  black. The dealer spins the wheel and at the same
  time rolls a small ball along the wheel in the
  opposite direction. The wheel is carefully
  balanced so that the ball is equally likely to
  land in any slot when the wheel slows. Gamblers
  can bet on various combinations of numbers and
  colours.
• a) What is the probability that the ball will
  land in any one slot?
• b) If you bet on red, you win if the ball
  lands in a red slot. What is the probability of
  winning?

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5. DISJOINT EVENTS

• Mutually exclusive (Disjoint) events The
  occurrence of any one event means that none of
  the others can occur at the same time.
• In the two coin experiment, the four possible
  outcomes are mutually exclusive.

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6. SOME BASIC RULES OF PROBABILITY

• Rule of addition If two events A and B are
  mutually exclusive, the probability of A or B
  occurring equals the sum of their respective
  probabilities. The rule is given in the
  following formula

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Lecture Exercise 2

• A couple decides to have three children. Of
  course, each child will be either a girl or a
  boy.
• (a) Using G for girl and B for boy, list all the
  possible sequences of sexes of their children in
  order of birth.
• (b) All outcomes in your sample space of part
  (a) are equally likely. What is the probability
  of each outcome?
• (c) What is the probability that the family will
  have exactly one girl among its three children?

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7. THE COMPLEMENT

• A Venn diagram illustrating the complement might
  look as

A
AC
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THE COMPLEMENT RULE

• The complement rule is used to determine the
  probability of an event occurring by subtracting
  the probability of the event NOT occurring from
  1.
• Let P(A) be the probability of event A and P(AC)
  be the event of not A (complement of A).
• The complement rule is given by the following
  equivalent relationships

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Lecture Exercise 3

• If probability of rain tomorrow is 0.8, what is
  the probability of no rain tomorrow?
• a) 0.8
• b) 0.2
• c) 0.0
• d) Cannot tell could be any number between
  0 and 1.

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NOTE

• The complement rule is very important in the
  study of probability. Often, it is more
  efficient to calculate the probability of an
  event happening by determining the probability of
  it is not happening and subtracting the result
  from 1.

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Joint Events

• Suppose that you toss a coin twice and count
  tails. Events of interest are
• A First toss is a tail B
  Second toss is a tail
• What is the probability that BOTH tosses are
  tails?

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8. INDEPENDENT EVENTS

• Definition Two events A and B are independent
  if the occurrence of one has no effect on the
  probability of the occurrence of the other.
• This is written symbolically as

• Important in chemistry since the independence of
  sampling is always assumed (A sample that is
  obtained for chemical analysis is independent of
  any other sample obtained for analysis).

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EXAMPLE

• A persons blood type is inherited from the
  parents and each them carries two genes for blood
  type. Each of these genes has probability 0.5 of
  being passed to the child.
• Suppose that both parents carry the O and A
  genes. If both parents contribute an O gene the
  child will have blood type O. Otherwise A.
• M event that the mother contributes her O gene.
• F event when the father contributes his O gene.
• What is the probability that the child will have
  O-type blood?

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Lecture Exercise 4

• A gambler knows that red and black are equally
  likely to occur on each spin of a roulette wheel.
  He observes five consecutive reds and bets
  heavily on red at the next spin. Asked why, he
  says that red is hot. And that the run of reds
  is likely to continue. Explain to the gambler
  what is wrong with this reasoning.
• After hearing your explanation the gambler moves
  to a poker game. He is dealt five straight cards.
  He remembers what you said and assumes that the
  next card dealt in the same hand is equally
  likely to be red or black. Is the gambler right
  or wrong? Why?

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2.1 RANDOM VARIABLE

• Definition A random variable is a numerical
  value determined by the outcome of a random
  experiment. (A quantity resulting from a random
  experiment that, by chance, can assume different
  values).
• EXAMPLE Consider a random experiment in which a
  coin is tossed two times. Let X be the number of
  heads. Let H represent the outcome of a head and
  T the outcome of a tail.

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EXAMPLE (Continued)

• The sample space for such an experiment will be
  TT, TH, HT, HH
• Thus the possible values of X (number of heads)
  are 0, 1, and 2.
• Note In this experiment, there are 4 possible
  outcomes in the sample space. Since they are all
  equally likely to occur, each outcome has a
  probability of 1/4 of occurring.

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EXAMPLE (Continued)

TT TH HT HH
0 1 1 2
X
Number of heads
Sample Space
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EXAMPLE (Continued)

• The outcome of zero heads occurred only once.
• The outcome of one head occurred three times.
• The outcome of two heads occurred three times.
• The outcome of three heads occurred only once.
• From the definition of a random variable, X as
  defined in this experiment, is a random variable.
  
• Its values are determined by the outcomes of the
  experiment.
• Note the random variable X is associating points
  in the sample space with points on the real line
  (0, 1, 2, 3).

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2.2 PROBABILITY DISTRIBUTIONS

• Definition A probability distribution is a list
  of all the outcomes of an experiment and their
  associated probabilities.
• The probability distribution for the random
  variable X (number of heads) in tossing a coin
  two times is shown next.

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Probability Distribution for the Three Tosses
of a Coin

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CHARACTERISTICS OF A PROBABILITY DISTRIBUTION

• The probability of an outcome must always be
  between 0 and 1.
• The sum of the probabilities is always 1.

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2.3 DISCRETE RANDOM VARIABLE

• Definition A discrete random variable is a
  variable that can assume only certain clearly
  separated values resulting from a count of some
  item of interest.
• EXAMPLE Let X be the number of heads when a
  coin is tossed 3 times. Here the values for X
  are 0, 1, 2 and 3.
• DISCRETE distributions (probability models)
• Bernoulli
• Binomial
• Poisson

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2.4 CONTINUOUS RANDOM VARIABLE

• Definition A continuous random variable is a
  variable that can assume one of an infinitely
  large number of values (within a continuum).
• EXAMPLE (a) Height, Weight, Length.
• (b) Dosage of a drug,
  Tablet dissolution time
• Continuous variables are represented by smooth,
  density curves, and the area under the curve is
  equal 1.
• CONTINUOUS distributions (probability models)
• Normal
• Students t
• ?2
• F

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2.5 THE MEAN OF A DISCRETE PROBABILITY
DISTRIBUTION

• The mean reports the central location of the
  data.
• The mean of a random variable is its long-run
  average.
• The mean of a probability distribution is also
  referred to as its expected value, E(X).
• The mean is a weighted average of the random
  variable, where probabilities are the weights.
  

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MEAN of a PROBABILITY DISTRIBUTION (continued)

• NOTE The mean of a discrete probability
  distribution is computed by the formula
• (6-1)
• where mX (Greek letter, mu) represents the
  mean and pi is the probability of the various
  outcomes X.

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2.6 The Variance Standard Deviation of a
Discrete Probability Distribution

• The variance measures the amount of spread or
  variation of a distribution.
• The variance of a discrete distribution is
  denoted by the Greek letter (sigma
  squared).
• The standard deviation, sX is obtained by taking
  the square root of .

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The Variance Standard Deviation (continued)

• NOTE The variance of a discrete probability
  distribution can be computed from the
  definitional formula
• or easier by the following computational formula

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Lecture Exercise 4

• In any probability distribution (model), which of
  the following is true?
• (a) All probabilities are equal
• (b) All probabilities are random
• (c) All probabilities are between 0 and 1
• (d) All probabilities are zero.

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Lecture Exercise 5

• Keno is a favourite game in casinos. Balls
  numbered 1 to 80 are tumbled in a machine as the
  bets are placed, then 20 of the balls are chosen
  at random. Players select numbers by marking a
  card. The simplest of the many wagers available
  is "Mark 1 Number". Your payoff is 3 on a 1 bet
  if the number you select is one of the chosen.
  Because 20 of 80 numbers are chosen, your
  probability of winning is 20/80, or 0.25.
• a) What is the probability distribution (the
  outcomes and their probabilities) of the payoff X
  on a single play?
• b) What is the mean payoff ?X?
• c) In the long run, how much does the casino
  keep/loose from each dollar bet?

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Lecture Exercise 6
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Summary

• Probability and expected values give us a
  language to describe randomness.
• Random phenomena are not chaotic, but instead
  randomness is a kind of order in the world, a
  long-run regularity. As opposed to either chaos
  or a determinism.
• Albert Einstein I cannot believe that God
  plays dice with the universe
• Statistical designs for producing data are
  founded on deliberate randomising.
• If you understand probability, Statistical
  Inference is stripped of mystery.