Welcome to Mathnasium’s Maths Tricks series. Today we are calculating the conditional probability of one event occurring, given that another has already occurred.

Two events are considered *dependent* if the occurrence of one event affects the probability of the other event. For example, if we draw a red marble from a bag of multicoloured marbles and do not replace it, the probabilities of drawing different coloured marbles change; so, drawing marbles of particular colours from a bag without replacing them are dependent events.

The probability of a dependent event, B, occurring given that another event, A, has already occurred is called a *conditional* probability and is denoted as: P(B | A). Conditional probabilities can change based on some event that has already occurred.

Follow the example below to find the conditional probability.

Example: A jar contains 8 quarters, 6 dimes, and 10 pennies. If a quarter is removed from the jar and not replaced, what is the probability of then randomly drawing another quarter?

Step 1: Identify the two dependent events.

The two dependent events are “drawing a quarter” and “drawing a quarter.” We want to find the probability of drawing a quarter given that we have already drawn one quarter from the jar:

P(quarter | quarter), or P(Q | Q).

The two dependent events are “drawing a quarter” and “drawing a quarter.” We want to find the probability of drawing a quarter given that we have already drawn one quarter from the jar:

P(quarter | quarter), or P(Q | Q).

Step 2: Find the probability of the first dependent event.

The probability of drawing a quarter is: P(Q) = 8⁄24 = 1⁄3.

The probability of drawing a quarter is: P(Q) = 8⁄24 = 1⁄3.

Step 3: Find the conditional probability of the second event occurring, given that the first event occurred.

The probability of drawing a quarter, given that you already drew a quarter is: P(Q | Q) = 7⁄23.

The probability of drawing a quarter, given that you already drew a quarter is: P(Q | Q) = 7⁄23.

Answer: 7⁄23.