What is the probability of drawing from a deck of 52 cards a…

What is the probability of drawing from a deck of $52$ cards a king, queen or jack of any suit, or a black card?

Explanation of Method

Let $F$ = "a face card (K, Q, J)" and $B$ = "a black card". These events overlap: the black face cards belong to both. So we use the addition rule for compatible events $P(F+B)=P(F)+P(B)-P(FB)$, and check by counting cards directly.

Step 1: The events

$F$: $3$ ranks $\times$ $4$ suits $=12$ cards, $P(F)=\frac{12}{52}$;
$B$: $26$ black cards, $P(B)=\frac{26}{52}$;
$FB$: black face cards, $3\times2=6$, $P(FB)=\frac6{52}$.

Step 2: Addition rule

\[ \boxed{P(F+B)=\frac{12}{52}+\frac{26}{52}-\frac6{52}=\frac{32}{52}=\frac8{13}\approx0.615.} \]

Step 3: Check by counting

Favourable cards: all $26$ black cards plus the $6$ red face cards $=32$ cards. $\frac{32}{52}=\frac8{13}$ ✓.

Step 4: A common mistake

$\frac{12}{52}+\frac{26}{52}=\frac{38}{52}$ counts the six black face cards twice.

Result

	Answer
Probability	$\frac{12+26-6}{52}=\frac8{13}\approx0.615$