# One card is drawn at random from a well shuffled deck of 52 cards. Are E and F independent if E : the card drawn is a king or queen F : the card drawn is a queen or jack?

This is a multi part question answered separately on Clay6.com

Toolbox:
• If A and B are independent events, $$P(A\cap\;B)=P(A)\;P(B)$$
Given a deck of 52 cards, there are 4 kings, 4 queens and 4 jacks, and the P(drawing a king) = P(drawin a queen) = P(drawing a jack) = $\large\frac{4}{52} = \frac{1}{13}$
Given E: the card drawn is a king or queen, P (E) = P(king) + P(queen) = $\large\frac{1}{13} + \frac{1}{13} = \frac{2}{13}$
Given F: the card drawn is a queen or a jack, P(F) = P(queen) + P(jack) = = $\large\frac{1}{13} + \frac{1}{13} = \frac{2}{13}$
If A and B are independent events, $$P(A\cap\;B)=P(A)\;P(B)$$
P (E $\cap F) = P (card drawn is a queen) =$\large\frac{4}{52} = \frac{1}{13}$P (E$\cap$F) = P(E)$\times$P(F) =$\large\frac{2}{13}$$\;\times$$\large\frac{2}{13} = \frac{4}{169} \;\neq \large\frac{1}{13}\$
Therefore E and F are NOT independent