It is known that 3 of the circuit boards from a production l

It is known that 3% of the circuit boards from a production line are defective. If a random sample of 120 circuit boards is taken from this production line, estimate the probability that the sample contains AT LEAST 2 defective boards.

a. Using the Poisson approximation

b. Using the Binomial approximation

Solution

a)

Using poisson approximation, we get a mean of

u = n p = 120(0.03) = 3.6

Now, note that

P(at least 2) = 1 - P(0) - P(1)

with Poisson distribution, using the formula P(x) = u^x e^(-u) / x!,

P(0) = 0.027323722
P(1) = 0.098365401

Thus,

P(at least 2) = 0.874310877 [ANSWER]
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b)

Now, for binomial approximation:

Note that P(at least 2) = 1 - P(at most 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    120      
p = the probability of a success =    0.03      
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   1   ) =    0.121829533
          
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.878170467 [ANSWER]