prob Probability PowerPoint notes Probability PowerPoint notes

Probability PowerPoint notes

1. prob 1
INTRODUCTION to
PROBABILITY

2. prob 2
BASIC CONCEPTS of
PROBABILITY
Experiment
Outcome
Sample Space
Discrete
Continuous
Event

3. prob 3
Interpretations of Probability
Mathematical
Empirical
Subjective

4. prob 4
MATHEMATICAL
PROBABILITY
P(E) =

numberofwaysaneventcanoccur
numberofpossibleoutcomes
......
.. .

5. prob 5
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

6. prob 6
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?

7. prob 7
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)

8. prob 8
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?

9. prob 9
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)

10. prob 10
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?

11. prob 11
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)

12. prob 12
NON MUTUALLY EXCLUSIVE
EVENTS
If A and B are not mutually exclusive,
P(A or B) = P(A) + P(B) - P(A and B)

13. prob 13
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 Resistance
high low
Scratch Rhigh 80 9
low 6 5

14. prob 14
RANDOM VARIABLES
Discrete
Continuous

15. prob 15
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

16. prob 16
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

17. prob 17
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.

18. prob 18
EXPECTED VALUE
Discrete Random Variable
E(X) = X
1P(X
1) + …. + X
nP(X
n)

19. prob 19
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?

20. prob 20
CUMULATIVE PROBABILITY
FUNCTIONS
For a discrete random variable X,
the cumulative function is:
F(X) = P(X < x)
= f(z) for all z < x

21. prob 21
PROBABILITY HISTOGRAMS
Equipment Failures
0
0.05
0.1
0.15
0.2
0.25
0.3
0123456789
EQUIPMENT FAILURES
IN ONE-MONTH
X f(x) F(X)
00.12 0.12
10.26 0.38
20.26 0.64
30.16 0.8
40.09 0.89
50.04 0.93
60.03 0.96
70.02 0.98
80.01 0.99
90.01 1
1

22. prob 22
Variance of a Discrete Probability
Distribution
Var(X) = [x - E(X)]
2
*f(x)

23. prob 23
SOME SPECIAL
DISCRETE RV’s
Binomial
Poisson
Geometric
Hypergeometric

24. prob 24
BINOMIAL
X = the number of successes in n
independent Bernoulli trials of an
experiment
f(x) =
nC
xp
x
(1-p)
n-x
for x = 0,1,2….n
f(x) = 0 otherwise

25. prob 25
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??

26. prob 26
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

27. prob 27
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

28. prob 28
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.

29. prob 29
GEOMETRIC DISTRIBUTION
X = # of trials until the first success
f(x) = p
x
(1-p)
n-x
for x = 0,1,2….n
f(x) = 0 otherwise

30. prob 30
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