Combined Probabilities

Many UCAT Decision Making probability questions involve combining the probabilities of two or more events. This lesson covers the AND rule (multiplication), the OR rule (addition), the critical distinction between independent and dependent events, and the use of tree diagrams for systematic probability calculation.

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The AND Rule (Multiplication)

When you need the probability that two events both occur, you multiply their probabilities.

For Independent Events

Two events are independent if the occurrence of one does not affect the probability of the other.

$P(A \text{ and } B) = P(A) \times P(B)$P(A and B)=P(A)×P(B)

Example: The probability of rain on Monday is 0.3, and the probability of rain on Tuesday is 0.4. If these are independent, the probability of rain on both days is:

$P(\text{rain Monday and Tuesday}) = 0.3 \times 0.4 = 0.12$P(rain Monday and Tuesday)=0.3×0.4=0.12

For Dependent Events

Two events are dependent if the occurrence of one changes the probability of the other.

$P(A \text{ and } B) = P(A) \times P(B \mid A)$P(A and B)=P(A)×P(B∣A)

Where P(B | A) means "the probability of B given that A has occurred."