Chapter Objectives

  • Understand and use the terminology of probability
  • Determine whether two events are mutually exclusive and whether two events are independent
  • Calculate probabilities using the addition rules and multiplication rules
  • Construct and interpret contingency tables
  • Construct and interpret Venn diagrams
  • Construct and interpret tree diagrams

Assignment

The Basics

  • The likelihood of an event is its probability
  • ‌Experiments are used to test the probability of events
  • When the experiment is attempted, it’s called a trial
  • The result of the trial is the outcome
  • All the possible results of an experiment is the sample space
  • In the case of flipping a coin …
    • Flipping the coin is the experiment
    • Each flip is a trial
    • The result of each flip is the outcome
    • The possible results are heads or tails, which is the sample space

Notation and Numbers

  • $P(A)$ means the probability of event $A$
  • For heads on a coin, $P(H)$ or $P(\text{heads})$
  • Probability is given as a percentage (typically in decimal form)
  • A probability of 0 means it definitely won’t happen
  • A probability of 1 means it definitely will happen
  • A probability of 0.5 means just as likely to happen as not

Theoretical vs Experimental Probability

  • ‌Theoretical is what you think will happen
\[\begin{align} P(A) &= \frac{\text{Number of outcomes with event }A}{\text{Total number of possible outcomes}} \end{align}\]
  • ‌Experimental is what actually happens, but only reliable after a large number of trials
\[\begin{align} P(A) &= \frac{\text{Number of times event } 𝐴 \text{ occurs}}{\text{Total number of trials}} \end{align}\]

AND vs OR

  • Roll a six-sided dice and $S=\{1, 2, 3, 4, 5, 6\}$

  • Let $A=\{2, 4, 6\}, 𝐵=\{4, 5, 6\}$

  • $P(A \text{ or } B)$ would be rolling anything from either $A$ or $B$

  • $P(A \text{ or } B)=\{2, 4, 5, 6\}=4/6$

  • $P(A \text{ and } B)$ means outcome must exist in both A and B
  • $P(A \text{ and } B)=\{4, 6\}=2/6$

Complements and Conditionals

  • A complement is what’s not in the set

  • $A=\{2, 4, 6\}$ so $A’=\{1, 3, 5\}$

  • $P(A|B)$ is conditional probability; the probability of A given B
  • $P(A|B)$ asks the probability of rolling an even, given that you rolled a number over 3
  • There are three numbers over 3, and two are even, so $P(A|B) = 2/3$

Mutual Exclusion

To find the probability of event $A$ given that event $B$ has already happened,

\[\begin{align} P(A | B) &= \frac{P(A \text{ and } B)}{P(B)} \end{align}\]