CollegePro Painting does home interior and exterior painting

\"College-Pro Painting does home interior and exterior painting. The company uses inexperienced painters that do not always do a high-quality job. It believes that its painting process can be described by a Poisson distribution with an average of 4.8 defects per 400 square feet of painting. a. What is the probability that a 400-square-foot
painted section will have fewer than 6 blemishes? b. What is the probability that six randomly sampled
sections of size 400 square feet will each have 7 or fewer blemishes?\"

The answer for b. that I got is wrong. Please show how you got the answer. Thanks.

Solution

a)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    4.8      
p = the probability of a success =    0      
x = our critical value of successes =    6      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   5   ) =    0.651006437
          
Which is also          
          
P(fewer than   6   ) =    0.651006437 [ANSWER]

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b)

Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    4.8      
          
x = the maximum number of successes =    7      
          
Then the cumulative probability is          
          
P(at most   7   ) =    0.886666171

Thus, if six sections have these,

P(six has at most 7 blemishes) = 0.886666171^6 = 0.485915592 [ANSWER]

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