prob 4
MATHEMATICAL PROBABILITY
P(E) =
number of ways an event can occur
number of possible outcomes
. . . . . .
. . .
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
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?
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)
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?
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)
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?
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)
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)
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 Resistancehigh low
Scratch R high 80 9low 6 5
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
prob 16
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
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.
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?
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
prob 21
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
prob 24
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
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??
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
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
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.
prob 29
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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