Probability

• Reading Assignment for Friday - Ruxton & Colegrave, Chapter 2, “Starting with a well-defined hypothesis” (NO QUIZ)
• Homework #4 - Coming soon…

Definition: A confidence interval is a range of values surrounding the sample estimate that is likely to contain the population parameter.

Definition: A 95% confidence interval provides a most-plausible range for a parameter. Values lying within the interval are most plausible, whereas those outside are less plausible, based on the data.

Definition: A random trial is a process or experiment that has two or more possible outcomes whose occurrence cannot be predicted with certainty.

Definition: An event is any potential subset of all the possible outcomes of a random trial.

Definition: The probability of an event is the proportion of times the event would occur if we repeated a random trial over and over again under the same conditions. Probability ranges between zero and one.

Definition: General addition rule \[ \mathrm{Pr[A \ or \ B]} = \mathrm{Pr[A]} + \mathrm{Pr[B]} - \mathrm{Pr[A \ and \ B]} \]

Definition: The conditional probability of an event is the probability of that event occurring given that another event has already occurred.

Definition: The conditional probability of an event B given that A occurred is \[ \mathrm{Pr[B \ | \ A]} = \frac{\mathrm{Pr[A \ and \ B]}}{\mathrm{Pr[A]}} \]

Definition: General multiplication rule \[ \mathrm{Pr[A \ and \ B]} = \mathrm{Pr[A]}\times\mathrm{Pr[B \ | \ A]} \]

Definition: The conditional probability of an event A given that B occurred is \[ \mathrm{Pr[B \ | \ A]} = \frac{\mathrm{Pr[A \ and \ B]}}{\mathrm{Pr[A]}} \]

Definition: General multiplication rule \[ \mathrm{Pr[A \ and \ B]} = \mathrm{Pr[B \ | \ A]}\times\mathrm{Pr[A]} \]

Definition: General multiplication rule \[ \mathrm{Pr[A \ and \ B]} = \mathrm{Pr[A \ | \ B]}\times\mathrm{Pr[B]} \]

Definition: Bayes Rule \[ \mathrm{Pr[B \ | \ A]} = \frac{\mathrm{Pr[A \ | \ B]}\times \mathrm{Pr[B]}}{\mathrm{Pr[A]}} \]

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