Transcript BASIC CONCEPTS

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?
Scratch R high
low
Shock Resistance
high
low
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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
VARIABLE
Possible 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 Variable
E(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)
= S f(z) for all z < x
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PROBABILITY HISTOGRAMS
EQUIPMENT FAILURES
IN ONE-MONTH
X
0
1
2
3
4
5
6
7
8
9
f(x)
0.12
0.26
0.26
0.16
0.09
0.04
0.03
0.02
0.01
0.01
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Equipment Failures
0.3
F(X)
0.12
0.38
0.64
0.8
0.89
0.93
0.96
0.98
0.99
1
0.25
0.2
0.15
0.1
0.05
0
0
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Variance of a Discrete Probability
Distribution
Var(X) =
S[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-l lx/x!
f(x) = 0
for x = 0,1,2,…...
otherwise
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EXAMPLES
Examples of the Poisson
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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 DISTRIBUTION
X = # of trials until the first success
f(x) = px(1-p)n-x
f(x) = 0
for x = 0,1,2….n
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
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