Twentyseven percent of likely US voters think that it is too

Twenty-seven percent of likely U.S. voters think that it is too easy to vote in the United States. You randomly select 12 likely U.S. voters. Find the probability that the number of likely U.S. voters who think that it is too easy to vote in the United States is (a) exactly three, (b) at least four, and (c) less than eight.

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 = 3 ) = ( 12 3 ) * ( 0.27^3) * ( 1 - 0.27 )^9
= 0.2549

b)

P( X < 4) = P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 12 3 ) * 0.27^3 * ( 1- 0.27 ) ^9 + ( 12 2 ) * 0.27^2 * ( 1- 0.27 ) ^10 + ( 12 1 ) * 0.27^1 * ( 1- 0.27 ) ^11 + ( 12 0 ) * 0.27^0 * ( 1- 0.27 ) ^12
= 0.5863
P( X > = 4 ) = 1 - P( X < 4) = 0.4137

c)
P( X < 8) = P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 12 7 ) * 0.27^7 * ( 1- 0.27 ) ^5 + ( 12 6 ) * 0.27^6 * ( 1- 0.27 ) ^6 + ( 12 5 ) * 0.27^5 * ( 1- 0.27 ) ^7 + ( 12 4 ) * 0.27^4 * ( 1- 0.27 ) ^8 + ( 12 3 ) * 0.27^3 * ( 1- 0.27 ) ^9 + ( 12 2 ) * 0.27^2 * ( 1- 0.27 ) ^10 + ( 12 1 ) * 0.27^1 * ( 1- 0.27 ) ^11 + ( 12 0 ) * 0.27^0 * ( 1- 0.27 ) ^12
= 0.9953