15 An experiment consists of tossing a fair coin three times

15. An experiment consists of tossing a fair coin three times and observing the outcomes. Let A be the event that at least one head is thrown, and let B be the event that at most two tails are thrown.

a. Find P(A)

b. Find P(B)

c. Are A and B independent events?

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

P(A) = ATLEAST ONE HEAD IS THROWN
P( X < 1) = P(X=0)   
= ( 3 0 ) * 0.5^0 * ( 1- 0.5 ) ^3   
= 0.125
P( X > = 1 ) = 1 - P( X < 1) = 0.875

P(A) = 0.875

P(B) = at most two tails are thrown
P( X < = 2) = P(X=2) + P(X=1) + P(X=0)
= ( 3 2 ) * 0.5^2 * ( 1- 0.5 ) ^1 + ( 3 1 ) * 0.5^1 * ( 1- 0.5 ) ^2 + ( 3 0 ) * 0.5^0 * ( 1- 0.5 ) ^3
= 0.875

P(B) = 0.875

c)
P( A and B) = P(A) * P(B) is the Independent events

P(A) + P(B) - P(A n B) = P( A u B)
0.875 + 0.875 - P( A n B ) = 1
P( A n B) = 1 - 0.875 - 0.875 =-0.75 which is out of bound

No, it is not indendent events