Assume Prboy Prgirl 05 Then if there are four children wha

Assume Pr(boy) = Pr(girl) = 0.5. Then if there are four children, what is the probability that: All four are the same gender Three are the same gender Two are boys and two are girls For problem #7 above, what is the mean of the distribution For problem #7 above, what is the variance of the distribution For problem #7 above, what is the standard deviation of the distribution

Solution

a)

Let = the number of boys.

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 =    0.5      
x = the number of successes =    4      
          
Thus, the probability is          
          
P (    4   ) =    0.0625

As this is also the probability that all are girls, then we multiply this by 2,

P(all same) = 0.0625*2 = 0.125 [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 =    0.5      
x = the number of successes =    3      
          
Thus, the probability is          
          
P (    3   ) =    0.25

As this is also the probability that 3 are girls, then we multiply this by 2,

P(all same) = 0.25*2 = 0.5 [ANSWER]

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c)

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 =    0.5      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.375 [ANSWER]

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8.

u = mean = np =    2 [ANSWER]

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9.
  
s^2 = variance = np(1-p) =    1 [ANSWER]

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10.

s = standard deviation = sqrt(np(1-p)) =    1 [ANSWER]