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INTRODUCTION to PROBABILITY

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BASIC CONCEPTS of PROBABILITY

Experiment Outcome Sample Space

Discrete Continuous

Event

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Interpretations of Probability Mathematical Empirical Subjective

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MATHEMATICAL PROBABILITY

P(E) =

number of ways an event can occur

number of possible outcomes

. . . . . .

. . .

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PROPERTIES

0 < P(E) < 1

P(E’) = 1 - P(E)

P(A or B) = P(A) + P(B) for two events, A and B, that do not intersect

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Example

A part is selected for testing. It could have been produced on any one of five cutting tools.

What is the probability that it was produced by the second tool?What is the probability that it was produced by the second or third tool?What is the probability that it was not produced by the second tool?

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INDEPENDENT EVENTS

Events A and B are independent events if the occurrence of A does not affect the probability of the occurrence of B.

If A and B are independent

P(A and B) = P(A)*P(B)

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Example

The probability that a lab specimen is contaminated is 0.05. Two samples are checked.

What is the probability that both are contaminated?

What is the probability that neither is contaminated?

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DEPENDENT EVENTS

Events A and B are dependent events if they are not independent.

If A and B are independent

P(A and B) = P(A)*P(B/A)

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Example

From a batch of 50 parts produced from a manufacturing run, two are selected at random without replacement?

What is the probability that the second part is defective given that the first part is defective?

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MUTUALLY EXCLUSIVE EVENTS

Events A and B are mutually exclusive if they cannot occur concurrently.

If A and B are mutually exclusive,

P(A or B) = P(A) + P(B)

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NON MUTUALLY EXCLUSIVE EVENTS

If A and B are not mutually exclusive,

P(A or B) = P(A) + P(B) - P(A and B)

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Example

Disks of polycarbonate plastic from a supplier are analyzed for scratch resistance and shock resistance. For a disk selected at random, what is the probability that it is high in shock or scratch resistance?

Shock Resistancehigh low

Scratch R high 80 9low 6 5

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RANDOM VARIABLES

Discrete Continuous

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DISCRETE RANDOM VARIABLES Maps the outcomes of an experiment to

real numbers The outcomes of the experiment are

countable.

Examples Equipment Failures in a One Month Period Number of Defective Castings

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CONTINUOUS RANDOM VARIABLEPossible outcomes of the experiment are represented by a continuous interval of numbers

Examples• force required to break a certain tensile

specimen• volume of a container• dimensions of a part

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Discrete RV Example

A part is selected for testing. It could have been produced on any one of five cutting tools. The experiment is to select one part.

• Define a random variable for the experiment.• Construct the probability distribution.• Construct a cumulative probability

distribution.

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EXPECTED VALUE

Discrete Random VariableE(X) = X1P(X1) + …. + XnP(Xn)

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Example

At a carnival, a game consists of rolling a fair die. You must play $4 to play this game. You roll one fair die, and win the amount showing (e.g... if you roll a one, you win one dollar.) If you were to play this game many times, what would be your expected winnings? Is this a fair game?

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CUMULATIVE PROBABILITY FUNCTIONS

For a discrete random variable X,the cumulative function is:

F(X) = P(X < x)= f(z) for all z < x

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PROBABILITY HISTOGRAMS

Equipment Failures

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9

EQUIPMENT FAILURES IN ONE-MONTH

X f(x) F(X)0 0.12 0.121 0.26 0.382 0.26 0.643 0.16 0.84 0.09 0.895 0.04 0.936 0.03 0.967 0.02 0.988 0.01 0.999 0.01 1

1

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Variance of a Discrete Probability Distribution

Var(X) = [x - E(X)]2*f(x)

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SOME SPECIAL DISCRETE RV’s

Binomial Poisson Geometric Hypergeometric

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BINOMIAL

X = the number of successes in n independent Bernoulli trials of an experiment

f(x) = nCxpx(1-p)n-x for x = 0,1,2….n

f(x) = 0 otherwise

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EXAMPLE

A manufacturer claims only 10% of his machines require repair within one year.

If 5 of 20 machines require repair, does this support or refute his claim??

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POISSON DISTRIBUTION

X = # of success in an interval of time, space, distance

f(x) = e-x/x! for x = 0,1,2,…...f(x) = 0 otherwise

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EXAMPLES

Examples of the Poisson• number of messages arriving for routing

through a switching center in a communications network

• number of imperfections in a bolt of cloth• number of arrivals at a retail outlet

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EXAMPLE of POISSON

The inspection of tin plates produced by a continuous electrolytic process. Assume that the number of imperfections spotted per minute is 0.2.

Find the probability of no more than one imperfection in a minute.

Find the probability of one imperfection in 3 minutes.

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GEOMETRIC DISTRIBUTIONX = # of trials until the first success

f(x) = px(1-p)n-x for x = 0,1,2….nf(x) = 0 otherwise

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Example of Geometric

The probability that a measuring device will show excessive drift is 0.05. A series of devices is tested. What is the probability that the 6th device will show excessive drift?Find the probability of the 1st drift on the 6th trail.

P(X=1) = (0.05)(0.95)5 = 0.039