Probability

• NO READING ASSIGNMENT FOR FRIDAY
• Exam #1: Friday, September 27 (in class)
  • No R on the exam!
  • Calculators allowed

Understanding uncertainty: Visualising probabilities

Ian Short & Mike Pearson

• 
  http://tinyurl.com/joja5dt

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.

Instead of events, we have values of random variables.

Parasitic wasps (yuck!): Two categorical variables - Parasitized or not; sex of laid egg (M or F)

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 \[ \mathrm{Pr[A \ and \ B]} = \mathrm{Pr[A]} \times \mathrm{Pr[B]} \]