2.3 Conditional Probability

Conditional Probability

A probability given that some event occurred is called conditional probability. Suppose $E$ is an event in a sample space $S$ with $P (E) > 0$ . The probability that event $A$ occurs once $E$ has occurred, the conditional probability of $A$ given $E$ , written $P (A | E)$ is defined in:

Suppose $S$ is an equiprobable space and $A$ and $E$ are events. Then $P (A | E) = \frac{P (A \cap E)}{P (E)} = \frac{| A \cap E |}{| E |}$

So basically, you make $E$ the new sample space, then take the probability of $A$ happening in $E$ .

Example

A fair coin is tossed 10 times, giving heads on each toss. What is the probability of a toss of heads on the eleventh toss?

Experiment: Tossing a fair coin 11 times
$E =$ The event that the first 10 tosses are heads
$A =$ The event that all heads appear

$| A \cap E | = 1$
$| E | = 2$
$∴ P (A | E) = \frac{1}{2}$