STAT 400

Wed. January 29th, 2020

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Sample Space & Events

In our example of a coin flip, where $S=\{H,T\}$, we can list out all possible events (all subsets of $S$):

• $\{H\}$
• $\{T\}$
• $\{H,T\}$
• $\empty$ (null set/null event)

We can do the same with our die roll example, $S=\{1,2,3,4,5,6\}$:

• $\empty$
• $\{1\}, \{2\}, \{3\}...\{6\}$
• $\{1,2\}, \{1,3\}... \{1,6\},\{2,3\}...\{6,6\}$
• $...$
• $\{1,2,3,4,5,6\}$

All of these events can be assigned probability values, as we previously discussed.

The key thing to remember about probability values is that they represents frequency over the long run – in other words, it’s the average amount of times that an event will occur if an experiment is repeated many times.

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Operations on Events

What is the probability that…

1. …event $A$ doesn’t happen?
2. …both event $A$ and event $B$ happen?
3. …either event $A$ or $B$ happens?

These questions represent new events that are generated from information about other events.

Definition 1. $A'$ or $A^C$ is called the complement of $A$ – the event where $A$ doesn’t happen.

Definition 2. $A\cap B$ is called the intersection of $A$ and $B$ – the event where both $A$ and $B$ happen.

Definition 3. $A\cup B$ is called the union of $A$ and $B$ – the event where either $A$ or $B$ happens.

Thus our three questions can be represented by $P(A')$, $P(A\cap B)$, and $P(A\cup B)$.

Because these operations give new events, we can chain multiple of them together: $P(A\cap B\cap C)$ is the probability that $A$, $B$, and $C$ all occur.

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Mutually Exclusive/Disjoint Events

Definition. When $A\cap B=\empty$, $A$ and $B$ are said to be mutually exclusive (or disjoint) events. This implies that it is not possible for $A$ and $B$ to both occur in a given experiment.

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Properties of Probability

Axioms of Probability

1. $P(A)\ge 0$ for all events $A$.
2. $P(S)=1$ for all sample spaces $S$.
3. If $A_1,A_2,...$ is an infinite sequence of disjoint events, then $P(A_1\cup A_2\cup ...)=\sum_{i=1}^\infin P(A_i)$.

We can derive all other properties of probability from these three axioms.

Properties

1. $P(\empty)=0$.
2. If $A_1,A_2...A_n$ is a finite sequence of disjoint events, then $P(A_1\cup A_2\cup ...\cup A_n)=\sum_{i=1}^n P(A_i)$. (Finite version of 3rd axiom)
3. $P(A)+P(A')=1$.
   • $A\cap A'=\empty$
   • $A\cup A'=S$
4. $P(A) \le 1$.
5. $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. (You need to subtract away the probability of events in $A$ AND $B$ so they don’t get double counted)
6. $P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(B\cap C)-P(A\cap C)+P(A\cap B\cap C)$. (Same reasoning as above)