Event A = the card is black
Event B = the card is a king
For the equation to hold true:
[tex]\begin{gathered} P(A\text{ and B) = }P(A|B)\times P(B) \\ P(\text{card is black and a king) = P(}card\text{ is black|card is king)}\times\text{ P(card is king)} \end{gathered}[/tex][tex]\begin{gathered} \text{There are 4 kings in a deck} \\ P(A|B)\text{ = }\frac{1}{4} \\ P(B)=\text{ }\frac{4}{52} \\ P(A\text{ and B) = }\frac{1}{4}\times\frac{4}{52}\text{ = }\frac{1}{52} \end{gathered}[/tex][tex]\begin{gathered} \text{There are 26 blacks in a deck} \\ P(B|A)\text{ = }\frac{1}{26} \\ P(A)=\text{ }\frac{26}{52} \\ P(A\text{ and B) = }\frac{1}{26}\times\frac{26}{52}\text{ = }\frac{1}{52} \end{gathered}[/tex][tex]\begin{gathered} \text{Hence, the equation holds true:} \\ \text{for }P(A\text{ and B) = }P(A|B)\times P(B) \\ \frac{1}{52}=\frac{1}{4}\times\frac{4}{52} \\ \end{gathered}[/tex]