Question

A card is drawn at random from a standard deck. Prove whether the following events are dependent or independent.
a) the events "heart" and "black"
b) the events "black" and "ace"
c) the events "under 10" and "red"

Answer 1

a)

There are 26 black cards and 13 heart so

P(black) = 26 / 52

P(heart) = 13 /52

Out of 52 cards no card is both black and heart so

P(black and heart) = 0

Since P(black and heart) is not equal to P(black)P(heart) so events  "heart" and "black" are dependent.

b)

There are 26 black cards and 3 are aces so

P(black) = 26 / 52 = 1/2

P(ace) = 4 /52 = 1/13

Out of 52 cards, 2 cards are black  and ace so

P(black and ace) = 2 /52 = 1 /26

Since

P(black and ace) =P(black)P(ace) = 1/26

So the events "black" and "ace" are independent.

c)

There are 26 red cards and 9*4 = 36 cards are under 10 (taking ace as one) so

P(red) = 26 / 52 = 1/2

P(under 10) = 36 /52 = 9/13

Out of 52 cards, 9*2 =18 cards are red and under 10 so

P(red and under 10) = 18 /52 = 9 /26

Since

P(red and under 10) =P(red)P(under 10) = 9/26

So the events "under 10" and "red" are independent.