How Independence Changes the Probability Multiplication Rule

Start with the general multiplication rule

For any two events $A$ and $B$,

$$P(A\cap B)=P(A)P(B\mid A).$$

The conditional probability $P(B\mid A)$ asks for the chance of $B$ after accounting for the fact that $A$ occurred. This formula works whether the events are dependent or independent.

Independence removes the conditional adjustment

Events are independent when learning that one occurred does not change the probability of the other. Then

$$P(B\mid A)=P(B),$$

so the rule simplifies to

$$P(A\cap B)=P(A)P(B).$$

For two fair dice, the first die does not change the second die. The probability of rolling a 2 followed by a 5 is therefore

$$\frac16\times\frac16=\frac1{36}.$$

Sampling design can create dependence

With replacement, returning a selected item restores the original population, so repeated selections can be independent. Without replacement, the first selection changes both the count of favorable outcomes and the total remaining count; the second probability is then conditional on the first result.

Always ask: Did the first event change the sample space or the mechanism for the second event? If yes, do not automatically multiply two unchanged marginal probabilities.

Independent is not the same as mutually exclusive

Mutually exclusive events cannot occur together, so $P(A\cap B)=0$. By contrast, independent events can occur together, and one event leaves the other's probability unchanged.

If two mutually exclusive events both have positive probability, they cannot be independent: once one occurs, the conditional probability of the other becomes zero.

Convert counts before multiplying

When a problem gives category counts, first compute each event probability as

$$\frac{\text{favorable outcomes}}{\text{total equally likely outcomes}}.$$

Then decide whether the events are independent or dependent and apply the appropriate multiplication rule. This separates arithmetic from the more important modeling decision.