R Pagano understanding Statistics in Behavioral Sciences 13t

R. Pagano, understanding Statistics in Behavioral Sciences 13th ed page 245 question #5 Using table B, if N = 12 and p=0.50 a) probalility of getting exactly 10P events?

B) probability of getting 11 or 12P events?

C) probability of getting at least 10P vents?

D) probability of getting a result as extreme as or more extreme thean 13P?

Solution

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

a)
P( X = 10 ) = ( 12 10 ) * ( 0.5^10) * ( 1 - 0.5 )^2
= 0.0161
b)
P( X = 11 ) = ( 12 11 ) * ( 0.5^11) * ( 1 - 0.5 )^1
= 0.0029
P( X = 12 ) = ( 12 12 ) * ( 0.5^12) * ( 1 - 0.5 )^0
= 0.0002
P( X = 11 OR X = 12) = 0.0161 + 0.0002 = 0.0163

c)
P( X = 10 ) = ( 12 10 ) * ( 0.5^10) * ( 1 - 0.5 )^2
= 0.0161
P( X >= 10 ) = P( X = 10 ) + P( X = 11 ) + P( X = 12) = 0.0161 + 0.0029 + 0.0002 = 0.0192