Calculate the probabilities and expected values for the foll

twelve items will be randomly selected from a population of Within the population there are fourteen red objects. A sample of twenty-eight objects A \'success\' is defined as a red object. Calculate the probabilities and expected values for the following events. E(x) Half the objects in the sample are red. None of the objects in the sample are red. Five of the objects in the sample are red. Probability Probability E(x) Probability- E(x)

Solution

a)

Note that the probability of x successes out of n trials is          
          
P(x) = C(N-K, n-x) C(K, x) / C(N, n)          
          
where          
N = population size =    28      
K = number of successes in the population =    14      
n = sample size =    12      
x = number of successes in the sample = 12/2 =    6      
          
Thus,          
          
P(   6   ) =    0.296432898 [ANSWER]

Here,

E(x) = N n / K = 6 [ANSWER]

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b)

Note that the probability of x successes out of n trials is          
          
P(x) = C(N-K, n-x) C(K, x) / C(N, n)          
          
where          
N = population size =    28      
K = number of successes in the population =    14      
n = sample size =    12      
x = number of successes in the sample =    0      
          
Thus,          
          
P(   0   ) =    2.99128*10^-6 [ANSWER]

Here,

E(x) = N n / K = 6 [ANSWER]

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c)

Note that the probability of x successes out of n trials is          
          
P(x) = C(N-K, n-x) C(K, x) / C(N, n)          
          
where          
N = population size =    28      
K = number of successes in the population =    14      
n = sample size =    12      
x = number of successes in the sample =    5      
          
Thus,          
          
P(   5   ) =    0.225853637 [ANSWER]

Here,

E(x) = N n / K = 6 [ANSWER]