4  Probability Models: Probability Measures

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.

  1. Compute \(\text{P}(A)\), where \(A\) is the event that the sum of the two dice is 4.



  2. Compute \(\text{P}(C)\), where \(C\) the event that the larger of the two rolls (or the common roll if a tie) is 3.




  3. Compute and interpret \(\text{P}(A\cap C)\). (Is it equal to the product of \(\text{P}(A)\) and \(\text{P}(C)\)?)




  4. 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.




  5. Construct a table and plot of \(\text{P}(X = x)\) for each possible value \(x\) of \(X\).




  6. 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 \(Y\).




  7. 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.

  1. \(\text{P}(R > Y)\).




  2. \(\text{P}(T < 30)\), where \(T = \min(R, Y)\).




  3. \(\text{P}(R > Y, W < 15)\), where \(W = |R - Y|\).




  4. \(\text{P}(R < 24)\).