Conditional probability
In a nutshell
If you have conditions that directly affect an event, the probabilities of outcomes of that event change. This is known as conditional probability.
Conditional probability
Definition
Conditional probability is defined as the likelihood of an event or outcome occurring, based on the occurrence of a previous event or outcome.
For instance, if you don't know whether a coin is biased or not, and in almost all previous flips the coin has landed heads, you would assume (and be correct in doing so) that the coin is more likely to land heads again.
Formula for conditional probability
For two events $A$ and $B$, to find the probability of $A$ given that event $B$ has happened, use the formula:
$P(A B) = \dfrac{P(A \cap B)}{P(B)}$
Note: Remember that $P(A\cap B)$ means the probability of $A$ intersect $B$ or both $A$ and $B$ occurring.
Example
A bag contains six red balls and ten blue balls. Mickey draws two balls without replacement. What is the probability of Mickey drawing $2$ red balls given that he drew a red ball first?
Calculate the probabilities needed in the formula.
$P(\text{(Red, Red)}\cap \text{(Red first)})=\dfrac{6}{16} \times \dfrac{5}{15}= \dfrac{30}{240}$
$P(\text{Red first})=\dfrac{6}{16}$
Substitute values into the formula.
$P(\text{Red,Red} \text{Red first}) = \dfrac{\dfrac{30}{240}}{\dfrac{6}{16}}$
Simplify.
$\dfrac{\dfrac{1}{8}}{\dfrac{3}{8}}=\dfrac{1}{8}\div\dfrac{3}{8}=\dfrac{1}{8}\times\dfrac{8}{3}=\underline{\dfrac{1}{3}}$
Note: To calculate $P(\text{Red, Red}\cap \text{Red first})$, use the fact that the only scenario where both events occur at the same time, happens when two reds are picked meaning that $P(\text{Red, Red}\cap \text{Red first})=P(\text{Red, Red})$.
Tree diagrams
You can visually display conditional probabilities on tree diagrams:
 $P(B)$
 Probability of B occurring.  $P(\overline{B})$  Probability of B not occurring.
 $P(AB)$
 Probability of A occurring, given B does.
 $P(A\cap B)$
 Probability of A and B both occurring.


Using the rules for finding probabilities from tree diagrams, you can obtain the formula above for conditional probability as following the top branch of the tree gives:
$P(B) \times P(AB) = P(A \cap B)$
Which rearranges to give:
$P(A B) = \dfrac{P(A \cap B)}{P(B)}$
Independence
Definition
Two events are independent if the outcome of one event does not affect the outcome of the other.
For two independent events $A$ and $B$:
$P(A \cap B) = P(A)P(B)$
This means that:
$P(AB) = \dfrac{P(A \cap B)}{P(B)} = \dfrac{P(A)P(B)}{P(B)} = P(A)$
This formula can be used to check for independence. If, and only if, this formula is satisfied, two events $A$ and $B$ independent.