The industry standards suggest that 20 of new vehicles requi

The industry standards suggest that 20% of new vehicles require warranty service within the first year. A dealer sold 20 Nissans yesterday. Use equation for Binomial Probability for part a) and Table II for part b) & c). Show work! What is the probability that none of these vehicles requires warranty service? Use the Binomial equation for P(X=0). What is the probability that exactly one of these vehicles requires warranty service? Determine the probability 3 or more of these vehicles require warranty service. Compute the mean and std. dev. of this probability distribution.

Solution

a)

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 =    20      
p = the probability of a success =    0.2      
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.011529215 [ANSWER]

******************

b)

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 =    20      
p = the probability of a success =    0.2      
x = the number of successes =    1      
          
Thus, the probability is          
          
P (    1   ) =    0.057646075 [ANSWER]

*************************

c)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.2      
x = our critical value of successes =    3      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   2   ) =    0.206084719
          
Thus, the probability of at least   3   successes is  
          
P(at least   3   ) =    0.793915281 [ANSWER]

**************************

d)

u = mean = np =    4   [ANSWER]  
          
s = standard deviation = sqrt(np(1-p)) =    1.788854382   [ANSWER]