Chapter9 intro to prob.ppt

8/10/2019 Chapter9 intro to prob.ppt

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Chapter outline

The idea of probability

Thinking about the randomness

Probability models

Assigning probabilities: finite number ofoutcomes

Assigning probabilities: intervals ofoutcomes

Normal probability models Random variables


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Some event where the outcomes isuncertain. Examples of such outcomeswould be the roll of a die, the amount ofrain that we get tomorrow, or who will

be the president of the United Sates inthe year 2004.

In each case, we dont know for surewhat will happen. For example, when we

toss a coin once, we dont know exactlywhat we will get (Head or Tail).


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Probability theory allows us to make somesense out of happening due to chance.

Example: If you flip a coin many times, about half

the time you get heads and the other half you gettails. In general, the more times you flip the coin,the closer the ratio of heads to tails comes to one.

Question: Why should this always be so? Answer: There is a mathematical rule governing

coin flippingit says that when you flip a coin, theoutcomes are about even between heads and tails.


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Thinking about randomness

A phenomenon is randomif each outcome isuncertain but there is nonetheless a regulardistribution of outcomes in a large number ofrepetitions.

Examples of random phenomena

The probability of any outcomes of a randomphenomenon is the proportionof times theoutcome would occur in a very long series ofrepetitions.


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Definitions

Sample space: the set of all possibleoutcomes. We denote S

Event: an outcome or a set of outcomesof a random phenomenon. An event is a

subset of the sample space.

Probabilityis the proportion of successof an event.

Probability model: a mathematicaldescription of a random phenomenonconsisting of two parts: S and a way ofassigning probabilities to events.


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Example 9.6 (P.232)

We roll two dice and record the up-

faces in order (first die, second die)

What is the sample space S?

What is the event A: roll a 5?


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Probability models

Example 9.6 (p.232): Rolling two dice

We roll two dice and record the up-faces in order

(first die, second die)

All possible outcomes

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Roll a 5 : {(1,4) (2,3) (3,2) (4,1)}


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Example 9.4 (P.229)

We roll two dice and count the spots on

the up-faces.

What is the sample space S?

What is the event B: I get an evennumber.?

What is the event C: I get an odd

number. ? What is the event D: I get a count less

than 4?


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Probability rules

Rule 1: For any event E, 0


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Assigning probabilities:

Case I: finite number of outcomes

Assign a probability to each individual

outcome.

These probabilities must be numbers

between 0 and 1 and must have sum 1.

Probability histogram is useful.


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Example 9.7 (P.233)

S={1,2,3,4,5,6,7,8,9}

Let X=first digit.

Probability model:

X 1 2 3 4 5 6 7 8 9 P 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9

P(X>=6)=?

P(X>6)=? P(5


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Assigning probabilities:

Case II: intervals of outcomes

Example: P(0.3


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Example 9.8 (page 235)

Exercise 9.9 (page 237)


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Random variables

Random variable: a variable whosevalue is a numerical outcome of a randomphenomenon. There are two kinds ofrandom variables corresponding to theways of assigning probabilities.

Discrete random variable: spread on the

number line discretely.Continuous random variable: interval


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Probability distributions

Probability distribution of a random

variable X: it tells what values X can take andhow to assign probabilities to those values.

Probability of discrete random variable: list of

the possible value of X and their probabilities

Probability of continuous random variable:

density curve.


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Random variables

Example: tossing a coin 4 times S={HHHH, HHHT,HHTH,,TTTT}, It has

16 possible outcomes.

Suppose that we are interested in number of

heads, then S={0,1,2,3,4} We can assign probabilities to each

outcome.

Example: Uniform distribution over[0,1]

S=(0,1)

We can assign probabilities over interval

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Chapter9 intro to prob.ppt