A new bakery succeeds 70% of the time. Suppose that 5 such businesses open in th

Question

A new bakery succeeds 70% of the time. Suppose that 5 such businesses open in the Western Sydney region. Assume that their successes are independent.

(a) Show that the probability that all five businesses succeed is 0.16807. You may use the binomial formula to answer this question.

(b) For questions (i) to (iii), use the binomial table below: (i) What is the probability that exactly 2 businesses succeed? (ii) What is the probability that at least 3 businesses succeed? (iii) What is the probability that all 5 businesses fail?

Binomial Probabilities Table P(X) P(-X) P(EX) P(eX) P(e-X) 0 0.99757 0 0.00243 0.00243 1 0.02835 0.03078 0.00243 0.96922 0.99757 2 0.1323 0.16308 0.03078 0.83692 0.96922 3 0.3087 0.47178 0.16308 0.52822 0.83692 4 0.36015 0.83193 0.47178 0.16807 0.52822 5 0.16807 1 0.83193 0 0.16807

Explanation / Answer

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 =    5      
p = the probability of a success =    0.7      
x = the number of successes =    5      
          
Thus, the probability is          
          
P (    5   ) =    0.16807 [ANSWER]

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b)

i.

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 =    5      
p = the probability of a success =    0.7      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.1323 [ANSWER]

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ii.

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 =    5      
p = the probability of a success =    0.7      
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   ) =    0.16308
          
Thus, the probability of at least   3   successes is  
          
P(at least   3   ) =    0.83692 [ANSWER]

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iii.

Hence, 0 succeeed.

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 =    5      
p = the probability of a success =    0.7      
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.00243 [ANSWER]

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c)

u = mean = np =    3.5 [ANSWER]
  
s = standard deviation = sqrt(np(1-p)) =    1.024695077 [ANSWER]

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