A fair coin is tossed 20 times and all tosses are independen

A fair coin is tossed 20 times and all tosses are independent (a) what is the true probability of obtaining exactly 10 heads? compute and compare the probability when approximated by an appropriate normal distribution

Solution

a)

EXACT:

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 =    20      
p = the probability of a success =    0.5      
x = the number of successes =    10      
          
Thus, the probability is          
          
P (    10   ) =    0.176197052 [ANSWER]

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NORMAL APPROXIMATION:

x1 = lower bound =    9.5      
x2 = upper bound =    10.5      
u = mean = np =    10      
          
s = standard deviation = sqrt(np(1-p)) =    2.236067977      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -0.223606798      
z2 = upper z score = (x2 - u) / s =    0.223606798      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.411531637      
P(z < z2) =    0.588468363      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.176936726   [ANSWER]

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As we can see, the results are close to each other.

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b)

EXACT:

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.5      
x = our critical value of successes =    7      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   7   ) =    0.131587982
          
Thus, the probability of at least   8   successes is  
          
P(more than   7   ) =    0.868412018 [ANSWER]

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NORMAL APPROXIMATION WITH CONTINUITY CORRECTION:

We first get the z score for the critical value:          
          
x = critical value =    7.5      
u = mean = np =    10      
          
s = standard deviation = sqrt(np(1-p)) =    2.236067977      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    -1.118033989      
          
Thus, the left tailed area is          
          
P(z <   -1.118033989   ) =    0.868223761 [ANSWER]

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NORMAL APPROXIMATION WITHOUT CONTINUITY CORRECTION:

We first get the z score for the critical value:          
          
x = critical value =    7      
u = mean = np =    10      
          
s = standard deviation = sqrt(np(1-p)) =    2.236067977      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    -1.341640786      
          
Thus, the left tailed area is          
          
P(z <   -1.341640786   ) =    0.910143753 [ANSWER]

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As we can see, the one with continuity correction is closer than that without continuity correction. [ANSWER]