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Dealing with Random Phenomena

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Examples:: Die, Coin, Cards, Survey, Experiments, Data Collection, ... Recall That... likely, probabilities for events are easy to find just by counting. ... – PowerPoint PPT presentation

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Title: 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.

7
Probability
  • Probabilities must be between 0 and 1, inclusive.
  • A probability of 0 indicates impossibility.
  • A probability of 1 indicates certainty.

8
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.)

9
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)

10
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.

11
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.

12
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.

13
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

14
  • 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.

16
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.

17
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).

18
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).)

19
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

21
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.

22
Tree Diagrams (cont.)
  • a nice example of a tree diagram and shows how we
    multiply the probabilities of the branches
    together

23
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.

24
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.
  • .

25
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.
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