4 Probability Models: Probability Measures

Probability models can be applied to any situation in which there are multiple potential outcomes and there is uncertainty about which outcome will occur.
- Outcomes, events, and random variables define what is possible.
- Probability measures determine how probable.
A probability measure, typically denoted $\text{P}$, assigns probabilities to events to quantify their relative likelihoods according to the assumptions of the model of the random phenomenon.
The probability of event $A$, computed according to probability measure $\text{P}(A)$, is denoted $\text{P}(A)$.
A valid probability measure $\text{P}$ must satisfy the following three logical consistency “axioms”.
- For any event $A$, $0 \le \text{P}(A) \le 1$.
- If $\Omega$ represents the sample space then $\text{P}(\Omega) = 1$.
- (Countable additivity.) If $A_1, A_2, A_3, \ldots$ are disjoint then \[ \text{P}(A_1 \cup A_2 \cup A_2 \cup \cdots) = \text{P}(A_1) + \text{P}(A_2) +\text{P}(A_3) + \cdots \]
Additional properties of a probability measure follow from the axioms
- Complement rule. For any event $A$, $\text{P}(A^c) = 1 - \text{P}(A)$.
- Subset rule. If $A \subseteq B$ then $\text{P}(A) \le \text{P}(B)$.
- Addition rule for two events. If $A$ and $B$ are any two events \[ \text{P}(A\cup B) = \text{P}(A) + \text{P}(B) - \text{P}(A \cap B) \]
- Law of total probability. If $C_1, C_2, C_3\ldots$ are disjoint events with $C_1\cup C_2 \cup C_3\cup \cdots =\Omega$, then \[ \text{P}(A) = \text{P}(A \cap C_1) + \text{P}(A \cap C_2) + \text{P}(A \cap C_3) + \cdots \]
A probability model (or probability space) is the collection of all outcomes, events, and random variables associated with a random phenomenon along with the probabilities of all events of interest under the assumptions of the model.
The axioms of a probability measure are minimal logical consistent requirements that ensure that probabilities of different events fit together in a valid, coherent way.
A single probability measure corresponds to a particular set of assumptions about the random phenomenon.
There can be many probability measures defined on a single sample space, each one corresponding to a different probability model for the random phenomenon.
Probabilities of events can change if the probability measure changes.

Example 4.1 Consider a Cal Poly student who frequently has blurry, bloodshot eyes, generally exhibits slow reaction time, always seems to have the munchies, and disappears at 4:20 each day. Which of the following events, $A$ or $B$, has a higher probability? (Assume the two probabilities are not equal.)

$A$: The student has a GPA above 3.0.
$B$: The student has a GPA above 3.0 and smokes marijuana regularly.

Example 4.2 Consider a single roll of a four-sided die, but suppose the die is weighted so that the outcomes are no longer equally likely. Suppose that the probability of event $\{2, 3\}$ is 0.5, of event $\{3, 4\}$ is 0.7, and of event $\{1, 2, 3\}$ is 0.6. In what particular way is the die weighted? That is, what is the probability of each the four possible outcomes?

Example 4.3 Consider again a single roll of a weighted four-sided die. Suppose that

Rolling a 1 is twice as likely as rolling a 4
Rolling a 2 is three times as likely as rolling a 4
Rolling a 3 is 1.5 times as likely as rolling a 4

In what particular way is the die weighted? That is, what is the probability of each the four possible outcomes? Compute the probability of these events: $\{2, 3\}$, $\{3, 4\}$, $\{1, 2, 3\}$.

4.1 Equally Likely Outcomes

For a sample space $\Omega$ with finitely many possible outcomes, assuming equally likely outcomes corresponds to a probability measure $\text{P}$ which satisfies \[ \text{P}(A) = \frac{|A|}{|\Omega|} = \frac{\text{number of outcomes in $A$}}{\text{number of outcomes in $\Omega$}} \qquad{\text{when outcomes are equally likely}} \]

Example 4.4 Roll a four-sided die twice, and record the result of each roll in sequence. One choice of probability measure $\textbf{P}$ corresponds to assuming that the die is fair, the rolls are independent, and the 16 possible outcomes are equally likely.

Compute $\text{P}(A)$, where $A$ is the event that the sum of the two dice is 4.
Compute $\text{P}(C)$, where $C$ the event that the larger of the two rolls (or the common roll if a tie) is 3.
Compute and interpret $\text{P}(A\cap C)$. (Is it equal to the product of $\text{P}(A)$ and $\text{P}(C)$?)
Let $X$ be the sum of the two dice. Compute $\text{P}(X = 4) \equiv \text{P}(\{X = 4\})$. Then interpret the probability both as a long relative frequency and as a relative likelihood.
Construct a table and plot of $\text{P}(X = x)$ for each possible value $x$ of $X$.
Let $Y$ be the larger of the two rolls (or the common value if both rolls are the same). Construct a table and plot of $\text{P}(Y = y)$ for each possible value $y$ of $Y$.
Compute $\text{P}(X = x, Y = y)$ for each possible $(x, y)$ pair.

4.2 Uniform Probability Measures

The continuous analog of equally likely outcomes is a uniform probability measure. When the sample space is uncountable, size is measured continuously (length, area, volume) rather that discretely (counting). \[ \text{P}(A) = \frac{|A|}{|\Omega|} = \frac{\text{size of } A}{\text{size of } \Omega} \qquad \text{if $\text{P}$ is a uniform probability measure} \]

Example 4.5 Regina and Cady plan to meet for lunch between noon and 1:00 but they are not sure of their arrival times. Recall the sample space from Example 3.2. Let $R$ be the random variable representing Regina’s arrival time (minutes after noon), and $Y$ for Cady. Assume a uniform probability measure $\text{P}$, which corresponds to assuming that they each arrive uniformly at random at a time between noon and 1:00, independently of each other. Compute and interpret the following probabilities.

$\text{P}(R > Y)$.
$\text{P}(T < 30)$, where $T = \min(R, Y)$.
$\text{P}(R > Y, W < 15)$, where $W = |R - Y|$.
$\text{P}(R < 24)$.