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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,

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.

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, .

- CAUTION outcomes are equally likely??Winning

Lottery?? 50-50?? - Rain Today?? Yes-No 50-50??

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.

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.

Probability

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

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.)

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)

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.

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.

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.

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

- 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

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.

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.

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).

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).)

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.

- 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

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.

Tree Diagrams (cont.)

- a nice example of a tree diagram and shows how we

multiply the probabilities of the branches

together

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.

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. - .

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.