Key Points

The probability of an event E occurring, given that event F has already occurred, is denoted by $P(E|F)$. It is calculated using the formula $P(E|F) = \frac{P(E \cap F)}{P(F)}$, provided that $P(F) \neq 0$.

For events E and F in a sample space S: 1. $0 \leq P(E|F) \leq 1$. 2. $P(S|F) = P(F|F) = 1$. 3. $P(E'|F) = 1 - P(E|F)$, where E' is the complement of E.

If A and B are any two events of a sample space S and F is an event such that $P(F) \neq 0$, then $P((A \cup B)|F) = P(A|F) + P(B|F) - P((A \cap B)|F)$. If A and B are disjoint, the last term is zero.

The probability of the simultaneous occurrence of two events E and F is given by $P(E \cap F) = P(E) \times P(F|E)$ where $P(E) \neq 0$. This can also be written as $P(E \cap F) = P(F) \times P(E|F)$ where $P(F) \neq 0$.

Two events E and F are independent if the occurrence of one does not affect the probability of the other. The mathematical condition for independence is $P(E \cap F) = P(E) \times P(F)$.

Independent events are defined by probability ($P(A \cap B) = P(A)P(B)$), while mutually exclusive events mean they cannot occur together ($A \cap B = \phi$). Two mutually exclusive events with non-zero probabilities cannot be independent.

If two events A and B are independent, then the following pairs are also independent: (A' and B), (A and B'), and (A' and B'). For example, $P(A' \cap B) = P(A') \times P(B)$.

If A and B are two independent events, the probability of the occurrence of at least one of them is given by $P(A \cup B) = 1 - P(A')P(B')$.

A set of events $E_1, E_2, \dots, E_n$ forms a partition of a sample space S if they are pairwise disjoint ($E_i \cap E_j = \phi$ for $i \neq j$), exhaustive ($E_1 \cup E_2 \cup \dots \cup E_n = S$), and each event has a non-zero probability ($P(E_i) > 0$).

If {E_1, E_2, \dots, E_n} is a partition of the sample space S, then for any event A, its total probability is the sum $P(A) = \sum_{j=1}^{n} P(E_j)P(A|E_j)$.

For a partition {E_1, E_2, \dots, E_n} and an event A, Bayes' theorem calculates a reverse conditional probability: $P(E_i|A) = \frac{P(E_i)P(A|E_i)}{\sum_{j=1}^{n} P(E_j)P(A|E_j)}$. It is used to update the probability of a cause given an observed effect.

A random variable is a real-valued function whose domain is the sample space of a random experiment. For example, in a toss of two coins, if X is the number of heads, its values can be $X(HH)=2$, $X(HT)=1$, $X(TH)=1$, and $X(TT)=0$.