Question

please answer both if u use hand writing please make sure is readable

From an ordinary deck of 52 cards, one card is drawn at random and removed from the deck. A 2^nd card is drawn next. Compute the chance that it will be a Spade!                                                              [3]

1. Two events A and B are such that A^c is independent of B. Show that B^c is independent of A!   [3]

Answer 1

1)chance that it will be a Spade =P(first card is spade and second card is spade)+P(first card is not spade and second card is spade)=(13/52)*(12/51)+(39/52)*(13/51)=1/4

2)

as Ac is independent of B

therefore P(Ac n B)=P(Ac)*P(B)

P(Ac n B)=(1-P(A))*(1-P(Bc)

P(B)-P(A n B) =1-P(A)-P(Bc)+P(A)*P(Bc) ...(1)   (cause P(B)=P(A n B)+P(Ac n B))

as P(B)+P(Bc)=1

therefore from equation (1)

P(A)-P(A n B)=P(A)*P(Bc)

P(A n Bc)=P(A)*P(Bc)     (As P(A)=P(A n B)+P(A n Bc))

hence A and Bc are independent