Question

One card is drawn at random from a well- shuffled deck of 52 cards. In which of the following cases are the events E and F independent?
E: the card drawn is a king or queen
F: the card drawn is a queen or jack

Answer 1

In a deck of 52 cards, 4 cards are kings, 4 cards are queens and 4 cards are jacks.
$P\left(E\right)=P\left(carddrawnisakingoraqueen\right)=P\left(King\right)+P\left(Queen\right)=\frac{4}{52}+\frac{4}{52}=\frac{8}{52}=\frac{2}{13}$
and P (F) = P (cards drawn is a queen or a jack)
$=P\left(Queen\right)+\left(Jack\right)=\frac{4}{52}+\frac{4}{52}=\frac{8}{52}=\frac{2}{13}$
Now, $P\left(E\right)\times P\left(F\right)=\frac{2}{13}\times \frac{2}{13}=\frac{4}{169}\ne P(E\cap F)\Rightarrow P(E\cap F)\ne P\left(E\right)P\left(F\right)$
Therefore, the events E and F are not independent.