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 $P$ , 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 $P (A)$ , is denoted $P (A)$ .
A valid probability measure $P$ must satisfy the following three logical consistency “axioms”.
- For any event $A$ , $0 \leq P (A) \leq 1$ .
- If $Ω$ represents the sample space then $P (Ω) = 1$ .
- (Countable additivity.) If $A_{1}, A_{2}, A_{3}, \dots$ are disjoint then $P (A_{1} \cup A_{2} \cup A_{2} \cup \dots) = P (A_{1}) + P (A_{2}) + P (A_{3}) + \dots$
Additional properties of a probability measure follow from the axioms
- Complement rule. For any event $A$ , $P (A^{c}) = 1 - P (A)$ .
- Subset rule. If $A \subseteq B$ then $P (A) \leq P (B)$ .
- Addition rule for two events. If $A$ and $B$ are any two events $P (A \cup B) = P (A) + P (B) - P (A \cap B)$
- Law of total probability. If $C_{1}, C_{2}, C_{3} \dots$ are disjoint events with $C_{1} \cup C_{2} \cup C_{3} \cup \dots = Ω$ , then $P (A) = P (A \cap C_{1}) + P (A \cap C_{2}) + P (A \cap C_{3}) + \dots$
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 $Ω$ with finitely many possible outcomes, assuming equally likely outcomes corresponds to a probability measure $P$ which satisfies $P (A) = \frac{| A |}{| Ω |} = \frac{number of outcomes in A}{number of outcomes in Ω} when outcomes are equally likely$

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

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

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 $P$ , 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.

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