Conditional Probability Notes

Conditional Probability

Definition

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

Notation

The probability of event B happening given that event A has already happened is denoted as $P(B|A)$.

Formula

The formula for conditional probability is:

$P(B|A) = \frac{P(A \cap B)}{P(A)}$

Where:

• $P(B|A)$ is the probability of event B given event A.
• $P(A \cap B)$ is the probability of both events A and B occurring (the intersection of A and B).
• $P(A)$ is the probability of event A occurring.

Explanation of Formula

• Given that event A has already happened, the sample space is reduced to the event A.
• We are interested in the part of event B that also lies within event A, which is the intersection of A and B.
• The probability of B given A is the ratio of the probability of the intersection of A and B to the probability of A.

Example: Marbles in a Bag

Consider a bag containing marbles of different colors and sizes. We have:

• Large black marbles: 5
• Large white marbles: 7
• Small black marbles: 4
• Small white marbles: 4

Total marbles: 20

Problem

If a marble is drawn from the bag and it is black, what is the probability that it is large?

Solution

We want to find $P(Large|Black)$, the probability that the marble is large given that it is black.

Using the formula:

$P(Large|Black) = \frac{P(Large \cap Black)}{P(Black)}$

• $P(Large \cap Black)$ is the probability of drawing a marble that is both large and black. There are 5 such marbles out of 20, so $P(Large \cap Black) = \frac{5}{20}$.
• $P(Black)$ is the probability of drawing a black marble. There are 9 black marbles out of 20, so $P(Black) = \frac{9}{20}$.

Therefore,

$P(Large|Black) = \frac{\frac{5}{20}}{\frac{9}{20}} = \frac{5}{20} \cdot \frac{20}{9} = \frac{5}{9}$

Intuitive Explanation

Since we know the marble is black, we only consider the black marbles. Out of the 9 black marbles, 5 are large. Thus, the probability that the marble is large given that it is black is $\frac{5}{9}$.

Key Points

• Conditional probability changes the sample space to the event that is known to have occurred.
• The formula $P(B|A) = \frac{P(A \cap B)}{P(A)}$ is used to calculate the conditional probability.
• Understanding conditional probability is crucial in many areas, including statistics, machine learning, and decision-making.