Key takeaway: Conditional probability is the probability that an event occurs given that another event is already known to have occurred, found by dividing the joint probability by the probability of the known event.
Grade 10 – GeometryS-CP.A.3Probability and Counting
Definition of Conditional Probability
Conditional probability updates a chance when you learn something new. Instead of looking at everyone, you look only at the cases where the known event happened, and ask what share of those cases also show the second event.
The condition shrinks the sample space. If you are told a student owns a bike, you ignore everyone without one, and the new total is the number of bike owners. Written as a formula, the answer divides the overlap of the two events by the size of the one you are given.
Formula and notation
For conditional probability, read \(P(A \mid B)\) as “the probability of \(A\) given \(B\)”:
\[ P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}, \qquad P(B) > 0 \]
With counts instead of probabilities, divide the number in both events by the number in \(B\).
Key facts
- The event after the bar is the condition and becomes the new whole.
- P(A given B) usually differs from P(B given A).
- A condition can raise, lower or leave unchanged the original probability.
- If the condition changes nothing, the two events are independent.
Conditional Probability example
Among 150 students, 90 own a bike, 72 own a skateboard, and 54 own both. A student is picked at random and owns a bike. What is the probability that this student also owns a skateboard? What if the student is known to own a skateboard, and the question is about a bike?
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Given a bike, the new total is the 90 bike owners. Of these, 54 own a skateboard.
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\(P(\text{skateboard} \mid \text{bike}) = \frac{54}{90} = 0.6\).
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Given a skateboard, the new total is 72: \(P(\text{bike} \mid \text{skateboard}) = \frac{54}{72} = 0.75\).
The probabilities are 0.6 and 0.75, so reversing the condition changes the answer.
Common mistake
Watch out
Asa ignores the condition in a conditional probability question and divides by the whole group: \(\frac{54}{150} = 0.36\) for the chance of a skateboard given a bike.
Why it is wrong
That number is the chance of owning both. Once you know the student owns a bike, the group shrinks to the 90 bike owners, so divide by 90, not 150. The condition is the new whole.
Related terms and lessons
- Geometry · Probability: Independence, Conditional Probability and Fair DecisionsLessonWorksheetQuiz
- Algebra 2 · Probability: Sample Spaces, Independence and Conditional ProbabilityLessonWorksheetQuiz
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