381 Probability-II (The Rules of Probability & Counting Rules) QSCI 381 – Lecture 8 (Larson and Farber, Sects )

381 Independent Events-I Two events are said to be if the occurrence of one does not affect the probability of the other, i.e. the probability of the event B is the same as the probability of the event B given A. Two events are independent therefore if: Note: Knowing that B has occurred doesn’t impact the probability of whether A will occur if A and B are independent.

381 Independent Events-II Which of these events are independent: Being male and playing football. In this class and enjoying tennis. Being in this class and knowing about statistics.

381 The Multiplication Rule-I The probability that two events A and B will occur in sequence is: If A and B are independent, then the multiplication rule becomes:

381 The Multiplication Rule-II (Example) Consider evaluating the probability of extinction of a species that consists of 10 sub-populations when the probability of an individual sub-population becoming extinct is 0.1. What is the probability of the species becoming extinct when the probability of extinction of one sub-population is independent of that of any of the others? What is the probability of extinction when the process leading to extinction is common to all sub-populations? Hint: Write down what you know (look for any events that are conditional on others).

381 Mutually Exclusive Events Two events A and B are if A and B cannot occur at the same time Questions to assess whether two events A and B are mutually exclusive: Can A and B occur at the same time? Do A and B have outcomes in common? Can you think of some mutually exclusive events?

381 The Additive Rule The probability that events A or B will occur is: If events A and B are mutually exclusive, then:

381 Using the Rules of Probability (Example) Blood Type OABABTotal Rh factor Positive Negative Total The above table is based on data for 409 randomly selected blood donors What is the probability that a donor has type O or type A blood? What is the probability that a donor has type B blood and is Rh-negative? What is the probability that a donor has type A blood and is Rh-negative? What is the probability that a donor is Rh-positive given he / she has blood type O?

381 Review of Concepts and Formulae

381 The Fundamental Counting Principle-I If one event can occur in m ways and a second event can occur in n ways, the number of ways in which the two events can occur in sequence is m x n.

381 The Fundamental Counting Principle-II (Examples) You are sampling salmon: there are four watersheds, three streams in each watershed and four species in each stream. How many ways to select one watershed, species, and stream? 4 x 3 x 4 = 48. Check this by listing them. You are ageing fish. The sample is 10 animals and the fish are numbered 1, How many ways are there to age 4 of the 10 fish (fish are aged once and once only). How many ways to select the first fish, the second fish….