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Probability of and


The probability of the and of two events, say $E$ and $F$, written $\p(E \and F)$, is the probability of both events happening. You might see equivalent notations $\p(EF)$, $\p(E ∩ F)$ and $\p(E,F)$ to mean the probability of and. How you calculate the probability of event $E$ and event $F$ happening depends on whether or not the events are "independent". In the same way that mutual exclusion makes it easy to calculate the probability of the or of events, independence is a property that makes it easy to calculate the and of events.

Independent Events

If events are independent then calculating the probability of and becomes simple multiplication:

Definition: Probability of and for independent events.

If two events: $E$, $F$ are independent then the probability of $E$ and $F$ occuring is: $$ \p(E \and F) = \p(E) \cdot \p(F) $$

This property applies regardless of how the probabilities of $E$ and $F$ were calculated and whether or not the events are mutually exclusive.

The independence principle extends to more than two events. For $n$ events $E_1, E_2, \dots E_n$ that are mutually independent of one another -- the independence equation also holds for all subsets of the events. $$ \p(E_1 \and E_2 \and \dots \and E_n) = \prod_{i=1}^n \p(E_i) $$

We can prove this equation by combining the definition of conditional probability and the definition of independence.

Proof: If $E$ is independent of $F$ then $\p(E \and F) = \p(E) \cdot \p(F)$

$$ \begin{align} \p(E|F) &= \frac{\p(E \and F)}{\p(F)} && \text{Definition of } \href{ ../../part1/cond_prob/}{\text{conditional probability}} \\ \p(E) &= \frac{\p(E \and F)}{\p(F)} && \text{Definition of } \href{ ../../part1/independence/}{\text{independence}} \\ \p(E \and F) &= \p(E) \cdot \p(F) && \text{Rearranging terms} \end{align} $$

See the chapter on independence to learn about when you can assume that two events are independent

Dependent Events

Events which are not independent are called dependent events. How can you calculate the and of dependent events? If your events are mutually exclusive you might be able to use a technique called DeMorgan's law, which we cover in a latter chapter. For the probability of and in dependent events there is a direct formula called the chain rule which can be directly derived from the definition of conditional probability:

Definition: The chain rule.

The formula in the definition of conditional probability can be re-arranged to derive a general way of calculating the probability of the and of any two events: $$ \p(E \and F) = \p(E | F) \cdot \p(F) $$

Of course there is nothing special about $E$ that says it should go first. Equivalently: $$ \p(E \and F) = \p(F \and E) = \p(F | E) \cdot \p(E) $$

We call this formula the "chain rule." Intuitively it states that the probability of observing events $E$ and $F$ is the probability of observing $F$, multiplied by the probability of observing $E$, given that you have observed $F$. It generalizes to more than two events: $$ \begin{align} \p(E_1 \and E_2 \and \dots \and E_n) = &\p(E_1) \cdot \p(E_2|E_1) \cdot \p(E_3 |E_1 \and E_2) \dots \\ &\p(E_n|E_1 \dots E_{n−1}) \end{align} $$