Conditional Probability

The probability of an event given that another event has already occurred.

Conditional probability: P(A|B) = P(A∩B) / P(B)

P(A)0.500

P(B)0.400

P(A∩B)0.200

P(A|B)0.500

P(B|A)0.400

P(A∪B)0.700

A and B are independent: P(A|B) = P(A)

P(A) = 0.50

P(B) = 0.40

P(A∩B) = 0.20

Definition

Conditional probability is the probability of an event occurring given that another event has already happened.

We write $P(A \mid B)$ — "the probability of $A$ given $B$ ".

The key idea: learning that $B$ occurred changes what we know, which can change how likely $A$ seems.

Picking from a bag

A bag contains 3 red and 2 blue marbles. You draw one and it's red (without looking first, someone tells you).

Before knowing: $P(\text{red}) = \tfrac{3}{5}$
Before knowing: $P(\text{blue}) = \tfrac{2}{5}$

Now you draw a second marble. Given the first was red, there are now 2 red and 2 blue left.

$P(\text{second is red} \mid \text{first was red}) = \tfrac{2}{4} = \tfrac{1}{2}$

The condition changed the probabilities.

Try it

In a class of 30 students, 18 play sport and 12 play an instrument. 6 play both. A student is chosen at random and plays sport. What is the probability they also play an instrument?

Solution

Of the 18 who play sport, 6 also play an instrument.

$P(\text{instrument} \mid \text{sport}) = \tfrac{6}{18} = \tfrac{1}{3}$

Related concepts

Statistics· Probability

ProbabilityA number between 0 and 1 measuring how likely an event is to occur.

Statistics· Distributions

Binomial DistributionThe probability distribution of the number of successes in a fixed number of independent yes/no trials.

Conditional Probability

Related concepts

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