Mathematics

Problem

Let

S

be a sample space and let

A, B\subseteq S

be events with

A\cup B = S

and

A\cap B = \emptyset

. a. (i) Prove that

A

and

B

are complementary events. b. (ii) Conversely, prove that if

B

is the complement of

A

, then

A\cup B = S

and

A\cap B = \emptyset

. c. (iii) In the experiment of drawing one card at random from a standard deck of 52 cards, let

A =

“the card drawn is red” and

B =

“the card drawn is black”. Verify that

A

and

B

satisfy the conditions proved in parts (i) and (ii). d. (iv) Calculate

P(A)

P(B)

and

P(A\cap B)

for this experiment.