Initial commit

Conditional Probability and Bayes Theorem.md

@@ -0,0 +1,40 @@
+#Math #Probability
+
+# Conditional Probability
+
+Conditional probability, or the probability of $A$ given $B$ is:
+
+$$
+P(A|B)
+$$
+
+Let's start with the probability of $P(A \cup B)$. We know that when $P(A | B)$, $B$ is given to be true. Therefore, we must divide the probably of $P(A \cup B)$ by $P(B)$.
+
+$$
+P(A | B) = \frac {P(A \cup B)} {P(B)}
+$$
+
+This defines $P(A | B)$ for events. When $P(A | B) = P(A)$, $A$ and $B$ are independent.
+
+# Bayes’ Theorem
+
+Let's start with the definitions of conditional probability:
+
+$$
+P(A | B) = \frac {P(A \cup B)} {P(B)} \\
+P(B | A) = \frac {P(A \cup B)} {P(A)}
+$$
+
+Rearrange the second equation to define $P(A \cup B)$:
+
+$$
+P(A \cup B) = P(A) P(B | A)
+$$
+
+Now substitute that equation into the first equation:
+
+$$
+P(A | B) = \frac {P(A) P(B | A)} {P(B)}
+$$
+
+The above equation is Bayes' Theorem for events.