A coin is tossed 100 times Use the normal curve approximatio

A coin is tossed 100 times. Use the normal curve approximation to find the probability of obtaining.** between 50 and 75 heads inclusive exactly 75 heads **set up above probabilities for using the normal distribution table.

Solution

Mean ( np ) = 50
Standard Deviation ( npq )= 100*0.5*0.5 = 5
Normal Distribution = Z= X- u / sd                   
To find P( X > a or X < b ) = P ( X > a ) + P( X < b)
a)
P(X < 50) = (50-50)/5
= 0/5= 0
= P ( Z <0) From Standard Normal Table
= 0.5
P(X > 75) = (75-50)/5
= 25/5 = 5
= P ( Z >5) From Standard Normal Table
= 0
P( X < 50 OR X > 75) = 0.5+0 = 0.5                  

P( 50 <= X < = 75 ) = 1 - 0.5 = 0.5
b)
Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

P( X = 75 ) = ( 100 75 ) * ( 0.5^75) * ( 1 - 0.5 )^25
= 0.00000019