Question

Decide if events A and B are independent using conditional probability.

(a)  Two dice are tossed. Let A = “sum of 8” and B = “both numbers are even.”

(b)  Select a single card from a standard deck. Let A = “a heart” and B = “an ace.”

(c)  A couple have known blood genotypes AB and BO. Let A = “their child has genotype BO” and B = “their child has blood type B.”

Answer 1

Using conditional probability,

P(A|B ) = P(A and B)/P(B)

If A and B are independent then, P(A and B) = P(A)*P(B)

So, P(A|B) = P(A)

Now, a) P(A) = 5/36 (since there are 5 ways of getting a sum of 8 when 2 die are rolled and there are total 36 possible outcomes)

P(A|B) = 3/9 (since there is three ways of getting a sum of 8 when both are even nos (4,4) (2,6), (6,2) and there total 9 ways of getting both even nos)

Since P(A|B) is not equal to P(A) these events are not independent.

b) P(B) = 4/52 or 1/13 (Since there are 4 aces in total )

P(B|A) = 1/13 (there is only one ace of hearts and total 13 hearts)

Here, P(B|A) = P(B)

So these events are independent.