Combined Conditional Probabilities

The fundamental rule for combined events is that the probability of both events occurring ($A$ and $B$) is found by multiplying the probability of the first by the conditional probability of the second.

General Formula: $P(A \cap B) = P(A) \times P(B|A)$ where $P(B|A)$ is the probability of $B$ adjusted for the fact that $A$ has already happened.

This rule can be extended to any number of events: $P(A \cap B \cap C) = P(A) \times P(B|A) \times P(C|A \cap B)$.

In 'without replacement' scenarios, the total number of items in the sample space (the denominator) decreases by $1$ for each subsequent event.

If the second event requires the same category as the first, the number of successful outcomes (the numerator) also decreases by $1$.

If the second event requires a different category, the numerator for that category remains the same, but the denominator still decreases.

Identify Dependence: Look for phrases like 'without replacement', 'takes two at once', or 'does not put it back'. These indicate that probabilities must be adjusted for the second event.

Fraction Management: When working with tree diagrams, avoid simplifying fractions until the final step. Keeping a common denominator makes it much easier to add different path probabilities together.

The 'At Least One' Shortcut: For 'at least one' questions, it is often faster to calculate $1 - P(\text{none})$. For example, $P(\text{at least one Red}) = 1 - P(\text{No Reds})$.

Sanity Check: Always ensure that the sum of all final outcome probabilities at the end of the tree equals exactly $1$.