Conditional Probability vs Independent Events

Conditional probability updates the chance of an event once you know another event has happened. It is written $P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}$ (the probability of $A$ given $B$), the probability of both divided by the probability of the given event. For example, if 40 percent of students take both Spanish and art and 50 percent take Spanish, then the chance of art given Spanish is $0.40 / 0.50 = 0.8$. Restricting attention to $B$ is what given means.

Events are independent when knowing that one happened tells you nothing about the other, so $P(A \mid B) = P(A)$ (the probability of $A$ given $B$ equals the probability of $A$). For independent events the multiplication rule gives $P(A \text{ and } B) = P(A) \times P(B)$. For example, two fair coin flips are independent, so the probability of two heads is $0.5 \times 0.5 = 0.25$. Independence is about influence, not overlap, so do not confuse it with mutually exclusive.