Page 117 - Six Sigma for electronics design and manufacturing

Six Sigma for Electronics Design and Manufacturing
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3.3.2 Examples of using the binomial distribution
Example 3.3
If the probability of failure of one part is 25%, what is the probability
that the next two parts out of four are also failures?
2
2
B(2, 4, 0.25) = (4!/2!2!)(0.25) (0.75) = 21%
Example 3.4
The probability of a failed part is 5%. If 20 parts are made from the
same machine, what is the average (expected value) and standard de-
viation of a failure? What is the probability that the first four parts
will fail?
E(x) = 20 · 0.05 = 1, Standard deviation = 2 0 · 0 .0 5 · 0 .9 5 = 0.975
Probability of the first four parts failing = (probability of part 1 fail
+ probability of part 2 fail + part 3 + part 4)
P(x = 1, 2, 3, 4) =
(1, 2, 3, 4; n, p) = 0.64 or 64%
3.3.3 The Poisson distribution
The Poisson distribution approximates the binomial distribution
when the number of trials (n) is large and the probability of each trial
(p) is small. In this case the variable , sometimes called the outcome
parameter of the distribution is equal to np. The formula for the Pois-
son distribution is as follows:
x
p(x, ) = e ( /x!) (3.7)
–
where x is the outcome during a specific time or region and is the av-
erage number of outcomes in the time interval or region and
Average = Variance = np =
Use of the Poisson distribution is more appropriate when each
event has an equal probability of failure, producing a “defect.” It is es-
pecially useful in complex production operations, where the possibili-
ties or opportunities of defects increase very rapidly, and the probabil-
ity of getting a single defect at a specific place or time is small. The
Poisson-distribution-based charts (C or U charts) should be used
when the area of opportunity or boundary of finding defects is kept
constant. Examples are:
Defects in a one-shift operation
Solder defects in one electronic product
Defect in one PCB