Conditional probability

Conditional probability is the probability of an event occurring, assuming that another event has already occurred. $P(B|A)$ is said as the probability of event B given A

Example

Let us say that 2 fair 6 sided dice are rolled and their face up values sum is 6. What is the probability that the face up value of the one dice is 2?

Solution

Let call the first dice $D_1$ and the second one $D_2$ . There are 5 ways for $D_1 + D_2 = 6$ and 2 of those ways (distinct) includes a 2. Therefore, our answer is $\frac {2} {5}$ .

Formula

The formula for conditional probability is $P(A \cap B) = P(A) \cdot P(B|A)$ where $P(B|A)$ represents the conditional probability. $P(B|A)$ is also said as the probability of event B occurring given event A occurs. $P(A \cap B)$ is the probability $P(A) \cdot P(B)$ . We can also represent $P(A|B)$ as $P(A|B) = \frac {P(A \cap B)} {P(A)}$

Different cases

What if $P(A) = 0$ ? We can easily fix this with a little bit of limits and integrals. We simply just want a random variable conditioned on a random

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Conditional probability

Contents

Example

Solution

Formula

Different cases