Probability

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Probability
It is remarkable that this science
(probability), which originated in the
consideration of games of chance, should have
become the most important object of human
knowledge. Pierre-Simon de Laplace
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Overview
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Rare Event Rule of Inferential Statistics

• If, under a given assumption, the probability of
  a particular observed event is extremely small,
  we conclude that the assumption is probably not
  correct.

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Fundamentals
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Some Definitions

• An event is any collection of results or outcomes
  of a procedure.
• A simple event is an outcome or an event that
  cannot be further broken down into simpler
  components.
• The sample space for a procedure consists of all
  possible simple events. That is, the sample space
  consists of all outcomes that cannot be broken
  down any further.

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Example

• What is the sample space for each procedure?
• Tossing a coin.
• Rolling a single fair die.

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Example (continued)

• List the possible outcomes for each event.
• Tossing a heads.
• Rolling an even number.

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Notation for Probabilities

• P denotes a probability
• A, B, and C denote specific events
• P(A) denotes the probability of event A
  occurring

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Rule 1 Relative Frequency Approximation of
Probability

• Conduct (or observe) a procedure, and count the
  number of times that event A actually occurs.
  Based on these actual results, P(A) is estimated
  as follows

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Rule 2 Classical Approach to Probability

• Assume that a given procedure has n different
  simple events and that each of those simple
  events has an equal chance of occurring. If event
  A can occur in s of these n ways, then

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Rounding Off Probabilities

• When expressing the value of a probability,
  either give the exact fraction or decimal or
  round off final decimal results to three
  significant digits.

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Example

• Each member of the class will toss a coin. Record
  the number of heads and tails observed.
• Estimate the probability of tossing a heads.
• Use the classical definition to calculate the
  probability of tossing a heads.

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Law of Large Numbers

• As a procedure is repeated again and again, the
  relative frequency probability (from Rule 1) of
  an event tends to approach the actual probability.

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Example

• Consider a couple that wants to have two
  children.
• What is the sample space for the gender of the
  two children?
• What is the probability of that both children are
  girls?
• What is the probability of having one girl and
  one boy?

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Rule 3 Subjective Probabilities

• P(A), the probability of event A, is estimated by
  using knowledge of the relevant circumstances.

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Simulation

• A simulation of a procedure is a process that
  behaves the same way as the procedure, so that
  similar results are produced.

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Properties of Probabilities

• The probability of an impossible event is 0.
• The probability of an event that is certain to
  occur is 1.
• For any event A, the probability of A is between
  0 and 1 inclusive. That is,

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Complementary Events

• The complement of event A, denoted by ,
  consists of all outcomes in which event A does
  not occur.

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Example

• Consider a couple that wants to have two
  children.
• Let A event that both children are girls, what is
  the complement of the event A?
• What is the probability of ?

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Addition Rule
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Compound Event

• A compound event is any event combining two or
  more simple events.
• Given two simple (or compound) events A and B,
  the following are compound events
• A and B
• A or B

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Notation for Addition Rule

• P(A or B) P(in a single trial, event A occurs
  or event B occurs or they both occur)

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Example

• Consider drawing a single card from an ordinary
  deck of cards.
• What is the probability of drawing a king or a
  queen?
• What is the probability of drawing a king or a
  heart?

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Mutually Exclusive

• Events A and B are disjoint (or mutually
  exclusive) if they cannot occur at the same time.

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Formal Addition Rule

• P(A or B) P(A) P(B) - P(A and B)
• If A and B are mutually exclusive, thenP(A or B)
  P(A) P(B)

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Example

• Use the data given in the table, which summarizes
  blood groups and Rh types for 100 typical people.
  These values may vary in different regions
  according to the ethnicity of the population.

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Example (cont.)

• If one person is randomly selected, find the
  probability of getting someone who is Rh.
• If one person is randomly selected, find the
  probability of getting someone who is group A or
  group B.
• If one person is randomly selected, find the
  probability of getting someone who is group B or
  Rh.

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Rule of Complementary Events

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Multiplication Rule
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Notation for Mulitiplication Rule

• P(A and B) P(event A occurs in a first trial
  and event B occurs in a second trial)

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Example

• Consider drawing two cards from an ordinary deck
  of cards, with replacement.
• What is the probability of drawing two kings?

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Example

• Consider drawing two cards from an ordinary deck
  of cards, without replacement.
• What is the probability of drawing two kings?

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Independence

• Two events A and B are independent if the
  occurrence of one does not affect the probability
  of the occurrence of the other. (Several events
  are similarly independent if the occurrence of
  any does not affect the probabilities of the
  occurrence of the others). If A and B are not
  independent, they are said to be dependent.

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Formal Multiplication Rule

• P(A and B) P(A) . P(BA)
• If A and B are independent, thenP(A and B)
  P(A) . P(B)

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Notation for Mulitiplication Rule

• P(BA) represents the probability of event B
  occurring after it is assumed that event A has
  already occurred. (We can read BA as B given A
  or as event B occurring after event A has
  already occurred.)

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Example

• Consider drawing two cards from an ordinary deck
  of cards, what is the probability of drawing two
  red cards if
• Replacement is allowed?
• Replacement is not allowed?

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Treating Dependent Events as Independent

• If a sample size is no more than 5 of the size
  of the population, treat the selections as being
  independent (even if the selections are made
  without replacement, so they are technically
  dependent).

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Example

• A geneticist develops a procedure for increasing
  the likelihood that offspring of fruit flies will
  be females. In an initial test, the parents are
  treated and the results consist of 20 females
  among 20 offspring. Assuming that the
  gender-selection procedure has no effect, find
  the probability of getting 20 females among 20
  offspring. Based on this result, is there strong
  evidence to support the claim that the procedure
  is effective in increasing the likelihood that
  the offspring will be females?

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Multiplication Rule Beyond the Basics
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Complements The Probability of At Least One

• At least one is equivalent to one or more.
• The complement of getting at least one item of a
  particular type is that you get no items of that
  type.
• To find the probability of at least one of
  something, calculate the probability of none,
  then subtract that result from 1. That is,
  P(at least one) 1 P(none)

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Example

• Consider a couple that wants to have three
  children.
• What is the probability of that all children are
  girls?
• What is the probability of having at least one
  boy?

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Example

• Consider drawing a single card from an ordinary
  deck of cards.
• What is the probability of drawing a diamond?
• What is the probability of drawing a diamond
  given that the card is red?

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Conditional Probability

• A conditional probability of an event is a
  probability obtained with the additional
  information that some other event has already
  occurred. P(BA) denotes the conditional
  probability of event B occurring, given that
  event A has already occurred, and it can be found
  by dividing the probability of events A and B
  both occurring by the probability of event A

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Example

• Use the data given in the table, which summarizes
  blood groups and Rh types for 100 typical people.
  These values may vary in different regions
  according to the ethnicity of the population.

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Example (cont.)

• If one person is randomly selected, find the
  probability of getting someone who is Rh.
• If one person is randomly selected, find the
  probability of getting someone who is group A
  given that the person is Rh.
• If one person is randomly selected, find the
  probability of getting someone who is Rh given
  that the person is group A.

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Counting
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Example

• Recall our couple that wants to have children.
  How many ways can the gender of the children be
  arranged if the couple has
• 2 children?
• 3 children?

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Fundamental Counting Rule

• For a sequence of two events in which the first
  event can occur m ways and the second event can
  occur n ways, the events together can occur a
  total of m . n ways.

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Example

• DNA (Deoxyribonucleic acid) is made up of
  nucleotides, and each nucleotide can contain on
  of these nitrogenous bases
• A (adenine),
• G (guanine),
• C (cytosine),
• T (thymine).
• If one of those four bases (A, G, C, T) must
  be selected three times to form a codon (a linear
  triplet), how many codons are possible?

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Example

• Recall that there are 20 different amino acids.
  If a protein consists of 100 amino acids, how
  many different proteins are possible?

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Factorial Rule

• A collection of n different items can be arranged
  in order n! different ways.
• Notation
• The factorial symbol ! Denotes the product of
  decreasing positive whole numbers.
• n! n(n 1) . . . 2 . 1
• 0! 1

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Example

• Evaluate the following
• 4!
• 5!
• 10!
• 20!

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Permutations Rule(When Items Are All Different)

• Requirements
• There are n different items available. (This rule
  does not apply if some of the items are identical
  to others.)
• We select r of the items (without replacement).
• We consider rearrangements of the same items to
  be different sequences.
• If the proceeding requirements are satisfied, the
  number of permutations of r items selected from n
  different available items (without replacement)
  is

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Example

• How many ways can we arrange students in the
  front row of class?

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Permutations Rule(When Some Items Are Identical
to Others)

• Requirements
• There are n different items available, and some
  items are identical to others.
• We select r of the items (without replacement).
• We consider rearrangements of the same items to
  be different sequences.
• If the proceeding requirements are satisfied, and
  if there are n1 alike, n2 alike, . . . , nk
  alike, the number of permutations of all items
  selected without replacement is

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Example

• Recall our couple that wants to have children.
  How many ways can the gender of the children be
  arranged if the couple
• Has 2 girls out of 3 children?
• Has 6 girls out of 10 children?

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Combinations Rule

• Requirements
• There are n different items available.
• We select r of the n items (without replacement).
• We consider rearrangements of the same items to
  be the same.
• If the proceeding requirements are satisfied, the
  number of combinations of r items selected from n
  different items is

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Example

• Recall our couple that wants to have children.
  Suppose they have six children, what is the
  probability that they have four girls?

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Risks and Odds
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Odds

• The actual odds against event A occurring are the
  ratio usually expressed in the form of ab (or
  a to b), where a and b are integers having no
  common factors.
• The actual odds in favor of event A are the
  reciprocal of the actual odds against that event.
  If the odds against A are ab, then the odds in
  favor of A are ba.
• The payoff odds against event A represent the
  ratio of net profit (if you win) to the amount
  bet. payoff odds against event A (net
  profit)(amount bet)

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Example

• A study collected data on the obesity status of
  children ages 5-6 years and the smoking status of
  the mother during pregnancy. For children
  whose mothers smoked throughout the pregnancy,
  what are the odds in favor of obesity?