Question on Conditional Probability and dependence

Question

If a card is chosen randomly from a standard deck, show that the events 'face card' and 'diamond' are independent.

I know to do this you need to show that $P(\text{Face Card}) = P(\text{Face Card} \mid \text{Diamond})$.

$P(F) =$ $\frac{(13)(4)}{52}=\frac{3}{13}$

P(F | D) means the probability of drawing a diamond given that you've drawn a face card right?

There are 3 face cards that are diamonds out of a total of 3*4 possible face cards, giving a probability of $\frac{3}{12}$. These are not equal though? Where did I make a mistake?

Answer 1

In general, $\Pr(A|B)$ is the probability that the event $A$ happens, given that the event $B$ happens. One reads it as "the probability of $A$, given $B$." The condition, the "given," is the second item, the item after the "$|$."

In particular, $\Pr(F|D)$ is the probability that the card is a face card, given it is a diamond. Now the calculation should go well!

Remark: The usual formal definition of $\Pr(A|B)$ is $$\Pr(A|B)=\frac{\Pr(A\cap B)}{\Pr(B)}$$ (if $\Pr(B)\ne 0$).

Answer 2

To show that two events are independent, we need to show (using your notation) $$P(F\cap D) = P(F) \times P(D)$$ In a standard deck, $P(F) = \frac {12}{52}$ and $P(D) = \frac {13}{52}$ and $P(F\cap D) = \frac{3}{52}$. You should be able to continue from here.