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?\"

Can someone explain why (.8867^6) is correct for b? Please explain how to find the (Lambda)(t) and the row for the correct number or if you have to substract, multiply or divdie something concerning part b.. 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

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Now, this is for ONE section. Thus, for 6 sections,

P(6 sections with 7 or fewer blemish) = 0.886666171^6
= 0.485915592 [ANSWER]

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All you need is to find where lambda = 4.8 is, then, for part b, x = 7. They meet at 0.8867. Then we raise that to 6 as we need 6 sections satisfying this condition.