A fair coin is flipped 400 times Determine the smallest posi

A fair coin is flipped 400 times. Determine the smallest positive integer x such that the probability that the number of heads is between 200 - x and 200 + x is at least 80%. Show your work and/or explain your answer.

Solution

u = mean = np =    200  
      
s = standard deviation = sqrt(np(1-p)) =    10  

As the middle area is          
          
Middle Area = P(x1<x<x2) =    0.8      
          
Then the left tailed area of the left endpoint is          
          
P(x<x1) = (1-P(x1<x<x2))/2 =    0.1      
          
Thus, the z score corresponding to the left endpoint, by table/technology, is          
          
z1 =    -1.281551566      
By symmetry,          
z2 =    1.281551566      
          
As          
          
u = mean =    200      
s = standard deviation =    10      
          
Then          
          
x1 = u + z1*s =    187.1844843      
x2 = u + z2*s =    212.8155157      

Thus,

200 + x = 212.8155157      

z = 12.8155157 = 13 [ANSWER]