Chapter 4 - Probability

4-1 Basic Concepts of Probability

• Probability plays a central role in the important statistical method of hypothesis testing

• An event is any collection of results/outcomes of a procedure

• A simple event is an outcome/event that cannot be further broken down into simpler components

• The sample space for a procedure consists of all possible simple events. That is, the sample space consists of all outcomes that cannot be broken down any further.

• 3 common approaches to finding the probability of an event:

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  1. relative frequency approximation of the probability
  2. the classical approach to probability (requires equally likely outcomes)
  3. subjective probabilities

• Law of large numbers: as a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability

• The complement of event A, denoted by A bar, consists of all outcomes in which event A does NOT occur

• If, under a given assumption, the probability of a particular observed event is very small and the observed event occurs significantly less than or significantly greater than what we typically expect with that assumption, we conclude that the assumption is probably not correct

• The actual odds against event A occurring are the ratio P(A bar) / P(A)

• The actual odds in favor of event A occurring are the ratio P(A) / P(A bar)

• The payoff odds against event A occurring are the ratio of net profit to the amount bet

4-2 Additional Rule and Multiplication Rule

• A compound event is any event combining 2 or more simple events
• P (A or B) = P(A) + P(B) - P(A and B)
• The addition rule is simplified when the events are disjoint or mutually exclusive (the events cannot occur at the same time)
• A bar represents the probability the event does not occur (complement)
• P(A and B) = P(A) * P(B|A)
• Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. If A and B are not independent, they are said to be dependent.
• Sampling with replacement: selections are independent events
• Sampling without replacement: selections are dependent events

4-3 Complements, Conditional Probability, and Bayes' Theorem

• A conditional probability of an event is a probability obtained with the additional information that some other event has already occurred
• Bayes' theorem is used to revise a probability value based on additional information that is later obtained
• A prior probability is an initial probability value originally obtained before any additional information is obtained
• A posterior probability is a probability value that has been revised by using additional information that is later obtained.

4-4 Counting

• The multiplication counting rule is used to find the total # of possibilities from some sequence of events
• Factorial Rule: The number of different arrangements (order matters) of n different items when all n of them are selected is n!
• Permutations of items are arrangements in which different sequences of the same items are counted SEPARATELY
• Combinations of items are arrangements in which different sequences of the same items are counted as being the SAME
• "Permutations Position" "Combinations Committee"

4-5 Probabilities Through Simulations

• A simulation of a procedure is a process that behaves the same way as the procedure, so that similar results are produced

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