Dealing with Random Phenomena

1
Dealing with Random Phenomena

• A random phenomenon is a situation in which we
  know what outcomes could happen, but we dont
  know which particular outcome did or will happen.
• When dealing with probability, we will be dealing
  with many random phenomena.
• Examples Die, Coin, Cards, Survey, Experiments,
  Data Collection,

2
Recall That

• For any random phenomenon, each trial generates
  an outcome.
• An event is any set or collection of outcomes.
• The collection of all possible outcomes is called
  the sample space, denoted S.

3
Events

• When outcomes are equally likely, probabilities
  for events are easy to find just by counting.
• When the k possible outcomes are equally likely,
  each has a probability of 1/k.
• For any event A that is made up of equally likely
  outcomes, .

4

• CAUTION outcomes are equally likely??Winning
  Lottery?? 50-50??
• Rain Today?? Yes-No 50-50??

5
Probability

• The probability of an event is its long-run
  relative frequency.
• For any random phenomenon, each attempt, or
  trial, generates an outcome. Something happens on
  each trial, and we call whatever happens the
  outcome.
• An event consists of a combination of outcomes.

6
Personal Probability

• We use the language of probability in everyday
  speech to express a degree of uncertainty without
  basing it on long-run relative frequencies.
• Such probabilities are called subjective or
  personal probabilities.
• Personal probabilities dont display the kind of
  consistency that we will need probabilities to
  have, so well stick with formally defined
  probabilities.

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Probability

• Probabilities must be between 0 and 1, inclusive.
• A probability of 0 indicates impossibility.
• A probability of 1 indicates certainty.

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Formal Probability

• Two requirements for a probability
• A probability is a number between 0 and 1.
• For any event A, 0 P(A) 1.
• Something has to happen rule
• The probability of the set of all possible
  outcomes of a trial must be 1.
• P(S) 1 (S represents the set of all possible
  outcomes.)

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Formal Probability (cont.)

• Complement Rule
• Definition The set of outcomes that are not in
  the event A is called the complement of A,
  denoted AC.
• The probability of an event occurring is 1 minus
  the probability that it doesnt occur.
• P(A) 1 P(AC)

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Formal Probability (cont.)

• Addition Rule
• Definition Events that have no outcomes in
  common (and, thus, cannot occur together) are
  called mutually exclusive.
• For two mutually exclusive events A and B, the
  probability that one or the other occurs is the
  sum of the probabilities of the two events.
• P(A or B) P(A) P(B), provided that A and B
  are mutually exclusive.

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Formal Probability (cont.)

• Multiplication Rule
• For two independent events A and B, the
  probability that both A and B occur is the
  product of the probabilities of the two events.
• P(A and B) P(A) x P(B), provided that A and B
  are independent.

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Putting the Rules to Work

• In most situations where we want to find a
  probability, well use the rules in combination.
• A good thing to remember is that it can be easier
  to work with the complement of the event were
  really interested in.

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The General Addition Rule

• General Addition Rule
• For any two events A and B,
• P(A or B) P(A) P(B) P(A and B).
• The following Venn diagram shows a situation in
  which we would use the general addition rule

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• A probability that takes into account a given
  condition is called a conditional probability.
• Example Chance Diabetes If Male,
• Example Chance Die Is 2 If Even,
• Example

15
It Depends

• To find the probability of the event B given the
  event A, we restrict our attention to the
  outcomes in A. We then find the fraction of those
  outcomes B that also occurred.
• Formally, .
• Note P(A) cannot equal 0, since we know that A
  has occurred.

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The General Multiplication Rule

• When two events A and B are independent, we can
  use the multiplication rule for independent
  events
• P(A and B) P(A) x P(B), provided that A and B
  are independent.
• However, when our events are not independent,
  this earlier multiplication rule does not work.
  Thus, we need the general multiplication rule.

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The General Multiplication Rule (cont.)

• Weve already encountered the general
  multiplication rule, but in the form of
  conditional probability. Rearranging the equation
  in the definition for conditional probability, we
  get
• General Multiplication Rule
• For any two events A and B,
• P(A and B) P(A) x P(BA) or
• P(A and B) P(B) x P(AB).

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Independence

• Recall that when we talk about independence of
  two events, we mean that the outcome of one event
  does not influence the probability of the other.
• With our new notation for conditional
  probabilities, we can now formalize this
  definition.
• Events A and B are independent whenever P(BA)
  P(B). (Equivalently, events A and B are
  independent whenever P(AB) P(A).)

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Independent ? Mutually Exclusive

• Mutually Exclusive events cannot be independent.
  Well, why not?
• Since we know that Mutually Exclusive events have
  no outcomes in common, knowing that one occurred
  means the other didnt. Thus, the probability of
  the second occurring changed based on our
  knowledge that the first occurred. It follows,
  then, that the two events are not independent.

20

• Example 4.56 Page 156
• Example 4.90 Page 167
• Example 4.96 Page 167
• Example 4.110 Page 174
• Example 4.116 Page 175
• Example
• Example

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Tree Diagrams

• The kind of picture that helps us think through
  conditional probabilities is called a tree
  diagram.
• A tree diagram shows sequences of events as paths
  that look like branches of a tree.
• Making a tree diagram for situations with
  conditional probabilities is consistent with our
  make a picture mantra.

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Tree Diagrams (cont.)

• a nice example of a tree diagram and shows how we
  multiply the probabilities of the branches
  together

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What Can Go Wrong?

• Beware of probabilities that dont add up to 1.
• Dont add probabilities of events if theyre not
  mutually exclusive.
• Dont multiply probabilities of events if theyre
  not independent.
• Dont confuse mutually exclusive and
  independentmutually exclusive events cant be
  independent.

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What Can Go Wrong?

• Dont use a simple probability rule where a
  general rule is appropriatedont assume that two
  events are independent or disjoint without
  checking that they are.
• .

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So What Do We Know?

• The addition rule for disjoint events can be
  generalized for any two events using the general
  addition rule.
• The multiplication rule for independent events
  can be generalized for any two events using the
  general multiplication rule.
• Conditional probabilities come from the general
  multiplication rule.
• Tree diagrams are helpful ways to display
  conditional events and probabilities.