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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.)
• 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
• 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