Question

A fair coin is tossed three times. The events A, B, E, F, M, N are described as given (1) A : head on third toss, B : head on first toss. Find P(A | B). (2) E : at least two heads, F : at most two heads. Find P(E | F). (3) M : at the most two tails, N : at least one tail. Find P(M | N).

Answer 1

Answer:

P(A  | B) = 1/2

P(E ∩F) =  3/7

P(M | N) =  6/7

Step-by-step explanation:

P(A  | B) = P(A ∩B)/P(B )

A ∩B = Head on 1st & third toss =>P(A ∩B) =  (1/2) *1 * (1/2) = 1/4

B = Head on third toss => P(B ) = 1 * 1 * (1/2) = 1/2

P(A  | B) = (1/4)/(1/2)  = 1/2

P(E | F)= P(E ∩ F)/P(F )

E ∩F = at least two heads  & at most two heads  => 2 heads

2 head   HHT , HTH , THH   = 1/8 + 1/8 + 1/8 = 3/8

P(E ∩F) =  3/8

F = at most two heads => not 3 heads = 1 - 1/8 = 7/8

P(E ∩F) = (3/8)/(7/8)  = 3/7

M : at the most two tails, N : at least one tail. Find P(M | N)

M ∩ N = one tail or two tail = 1  - (all tail , no tail) = 1 - 2/8  = 6/8

N : at least one tail = 1 - No tail = 1 - 1/8 = 7/8

P(M | N) = (6/8)/(7/8) = 6/7