What is the probability that at least one of the three trusts the government? 5.

Question

What is the probability that at least one of the three trusts the government? 5.36 Marketing and Consumer Behavior Twinkies are an American tradition, sold for over 83 years, originating in a Chicago bakery, and forever linked to legal lingo. (You may have read about the "Twinkie defense," associated with a 1979 murder trial in San Francisco.) Approximately 12% o f households in the United States buy Twinkies.^7 Suppose four households are selected at random and let Y be the total number of households that buy Twinkies. Construct the probability distribution for Y. Write a Solution Trail for this problem. What is the probability that at least one household buys Twinkies? Suppose at least two households buy Twinkies. What is the probability that all four households buy Twinkies? 5.37 Manufacturing and Product Development Staples has

Explanation / Answer

A)

Note that the probability of x successes out of n trials is          
          
P(Y) = nCY p^Y (1 - p)^(n - Y)          
          
where          
          
n = number of trials =    4      
p = the probability of a success =    0.12      
Y = the number of successes

Hence,

P(Y) = 4CY 0.12^Y (1 - 0.12)^(4 - Y)   [ANSWER]

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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 =    4      
p = the probability of a success =    0.12      
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.59969536

Thus, P(at least one) = 1 - P(0) =   0.40030464 [ANSWER]

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c)

Note that

P(Y=4|Y>=2) = P(Y=4)/P(Y>=2)

FOR P(Y=4):

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 =    4      
p = the probability of a success =    0.12      
x = the number of successes =    4      
          
Thus, the probability is          
          
P (    4   ) =    0.00020736
      
FOR P(Y>=2):

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 =    4      
p = the probability of a success =    0.12      
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   ) =    0.92680192
          
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.07319808
      
Hence,

P(Y=4|Y>=2) = P(Y=4)/P(Y>=2) = 0.00020736/0.07319808 = 0.002832861 [ANSWER]

      

          

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