Question

If 5 cards are drawn from a shuffled deck, verify that the events "drew 3 diamonds" and "drew 2 clubs" are not independent.

Answer 1

P(3 diamonds) =P(select 3 diamond from13 and 2 cards from rest of 39)=(₁₃C₃)*(₃₉C₂)/(₅₂C₅)

=286*741/2598960 =211926/2598960

P(2 clubs) =P(select 2 clubs from13 and 3 cards from rest of 39)=(₁₃C₂)*(₃₉C₃)/(₅₂C₅)

=78*9139/2598960 =712842/2598960

P(3 diamonds and 2 clubs)=(₁₃C₃)*(₁₃C₂)/(₅₂C₅)=(286*78)/2598960 =22308/2598960

since P(3 diamonds) *P(2 clubs) =(211926/2598960)*(712842/2598960)=0.0224 is not equal to

P(3 diamonds and 2 clubs) =22308/2598960 =0.0086

since they both are not equal , events "drew 3 diamonds" and "drew 2 clubs" are not independent.