Probability (Part 2)

• Solutions for Homeworks #1-3 are on Blackboard!!!
• Exam #1
  • Friday, September 27, 1:00 pm (HERE)
  • NO R! Bring a calculator!
  • Material covered:
    • Whitlock & Schluter, Chapters 1-4
    • Ruxton & Colegrave, Chapter 1
  • How to study:
    • Do Practice Problems
    • Read chapter summaries
    • Come to lecture next week!

• Format of next week:
  • I will be out of town at a conference M-Th
  • NO LABS
  • Lecture will be held by your TAs (Kathryn and Sam)
    • Monday: Overview of material
    • Wednesday: Q&A study session (You must bring questions to Wednesday or TAs will dismiss early)
  • TA office hour: Tuesday, 9:30 am, ISC 3252
  • I will be monitoring Slack, so you can also post questions there

Commonly confused!

Definition: Two events are mutually exclusive if they cannot both occur at the same time. \[ \mathrm{Pr[A \ and \ B]} = 0 \]

Definition: Two events are independent if the occurrence of one does not inform us about the probability that the second will occur. \[ \mathrm{Pr[B \ | \ A]} = \mathrm{Pr[B]} \]

These two conditions simplify the general additive and multiplicative rules:

If two events are mutually exclusive, then \[ \mathrm{Pr[A \ or \ B]} = \mathrm{Pr[A]} + \mathrm{Pr[B]} \]

If two events are independent, then \[ \mathrm{Pr[A \ and \ B]} = \mathrm{Pr[A]} \times \mathrm{Pr[B]} \]

Definition: The probability of an event not occurring is one minus the probability that it occurs. \[ \mathrm{Pr[{\it not}\ A]} = 1-\mbox{Pr[A]} \]

Definition: The law of total probability is given by \[ \begin{align*} \mathrm{Pr[A]} & = \sum_{B\ \mathrm{in} \ \mathcal{M}}\mathrm{Pr[A \ and \ B]} \\ & = \sum_{B\ \mathrm{in} \ \mathcal{M}} \mathrm{Pr[B]}\ \mathrm{Pr[A\ | \ B]}, \end{align*} \] where \( \mathcal{M} \) is a set of mutually exclusive events such that \[ \sum_{B\ \mathrm{in} \ \mathcal{M}}\mathrm{Pr[B]} = 1 \]

Definition: A probability distribution is a list of the probabilities of all mutually exclusive outcomes of a random trial.

Compare to:

Definition: A probability distribution (or relative frequency distribution) is a list of the probabilities of all values of a random variable in a sample or population.

Probability densities

1. Write out the probability that you are being asked to find. Is it a conditional probability? AND? OR?
2. Identify probabilities that you are given (again, are these conditionals? ANDs? ORs?)
3. Draw a probability tree (if appropriate)