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Apr 26, 2023 | Woodmore

Welcome to Mathnasium’s Math 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 multicolored marbles and do not replace it, the probabilities of drawing different colored marbles change; so, drawing marbles of particular colors 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.

**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).

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

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}.

**Answer:** ^{7}⁄_{23}.

Now, with this strategy, you are ready to calculate conditional probabilities. Click here for more practice problems, then check your answers here.

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