Please show your work Work on problem number 14 ad on page 6

Please show your work.

Work on problem number 14 (a-d) on page 6-19

Solution

14.

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 < = 4) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 20 4 ) * 0.5^4 * ( 1- 0.5 ) ^16 + ( 20 3 ) * 0.5^3 * ( 1- 0.5 ) ^17
   + ( 20 2 ) * 0.5^2 * ( 1- 0.5 ) ^18 + ( 20 1 ) * 0.5^1 * ( 1- 0.5 ) ^19 +
   ( 20 0 ) * 0.5^0 * ( 1- 0.5 ) ^20 +   
= 0.0059
b)
P( X = 6 ) = ( 20 6 ) * ( 0.5^6) * ( 1 - 0.5 )^14
= 0.037
c)
P( X > 5) = 1 - P ( X <=5) = 1 -0.0207 = 0.9793

d)
P( X < 3) = P(X=2) + P(X=1) + P(X=0)
= ( 20 2 ) * 0.5^2 * ( 1- 0.5 ) ^18 + ( 20 1 ) * 0.5^1 * ( 1- 0.5 ) ^19 + ( 20 0 ) * 0.5^0 * ( 1- 0.5 ) ^20
= 0.0002
P( X < = 7) = P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P( X < 3)
= ( 20 7 ) * 0.5^7 * ( 1- 0.5 ) ^13 + ( 20 6 ) * 0.5^6 * ( 1- 0.5 ) ^14 + ( 20 5 ) * 0.5^5 * ( 1- 0.5 ) ^15 + ( 20 4 ) * 0.5^4 * ( 1- 0.5 ) ^16 + ( 20 3 ) * 0.5^3 * ( 1- 0.5 ) ^17 + 0.0002
= 0.1316

P(3<=X<=7) = P( X < = 7) - P( X < 3) = 0.1316 - 0.0002 = 0.1314