Question

An urn contains eight balls labeled 1,2,···,8. One ball is drawn at random.

Define A={1,2,3,4} B={1,2,5,6}

C={1,2,7,8} D={1,3,5,7}

P(A)P(B)P(C)P(D) =P(A∩B)P(A∩C)P(A∩D)P(A∩B∩D) P(1) =

2. Are events A,B, and D(mutually) independent?

Answer 1

a) We are given here that:
A = { 1, 2, 3, 4}

Therefore, P(A) = 4/8 = 0.5 is the required probability here.
Similarly as all B, C and D have got 4 outcomes within them,

Therefore, P(B) = P(C) = P(D) = 4/8 = 0.5

Therefore, P(A) = P(B) = P(C) = P(D) = 0.5

P(A B) = P({ A and B} ) = P({1, 2}) = 2/8 = 0.25
Therefore, 0.25 is the required probability here.

P(A C) = P({ A and C} ) = P({1, 2}) = 2/8 = 0.25
Therefore, 0.25 is the required probability here.

P(A D) = P({ A and D} ) = P({1, 3}) = 2/8 = 0.25
Therefore, 0.25 is the required probability here.

P(A B D) = P({ A and B and D} ) = P({1}) = 1/8 = 0.125
Therefore, 0.125 is the required probability here.

Similarly, P(1) = 1/8 = 0.125
Therefore, 0.125 is the required probability here.

2) P({ A and B and D} ) = 0.125 as computed above.
P(A)(B)P(D) = 0.5*0.5*0.5 = 0.125 which is equal to P({ A and B and D} )

Therefore, A B and D are mutually independent events here.