1.The Denver Post reported that 80% of all new products introduced in grocery st

Question

1.The Denver Post reported that 80% of all new products introduced in grocery stores fail within 2 years. If a grocery store chain introduces 36 new products, what is the probability that within 2 years exactly 30 will fail?

2.You are trying to determine if you should accept a shipment of balloons for a local party store. About 5% of all balloons which are shipped are defective (they have a hole and cannot be blown up). You are instructed to accept the shipment if no more than 20 balloons out of the 350 you inspect are defective. What is the probability that you accept the shipment? (In other words, what is the probability that, at the most, you had 20 defective balloons?)

Explanation / Answer

1.

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    36      
p = the probability of a success =    0.8      
x = the number of successes =    30      
          
Thus, the probability is          
          
P (    30   ) =    0.154319981 [ANSWER]

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2.

Here, we can use the normal approximation to the binomial.

We first get the z score for the critical value:          
          
x = critical value =    20.5      
u = mean = np =    17.5      
          
s = standard deviation = sqrt(np(1-p)) =    4.077376608      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    0.735767207      
          
Thus, the left tailed area is          
          
P(z <   0.735767207   ) =    0.769063809 [ANSWER]

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