Combined Probability

What Is Combined Probability?

Combined probability involves calculating the chance of two or more events happening in a row. You can work these out even without a tree diagram using just two simple rules:

• "And" means multiply
• "Or" means add

These rules are based on the following formulas:

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

$$P(A \text{ or } B) = P(A) + P(B)\quad\text{(if events A and B are mutually exclusive)}$$

Also remember:

$$P(\text{not A}) = 1 - P(A)$$

Example

A bag contains 4 green marbles and 6 blue marbles. A marble is taken at random, noted, then replaced. Another marble is then chosen.

(a) Find the probability that both marbles are blue.

We're looking for:

$$P(\text{blue and blue}) = P(\text{blue}) \times P(\text{blue}) \\ P(\text{blue}) = \frac{6}{10} = \frac{3}{5} \\ P(\text{blue and blue}) = \frac{3}{5} \times \frac{3}{5} = \frac{9}{25}$$

Final Answer: $\frac{9}{25}$