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Probability-II (The Rules of Probability & Counting Rules)

Probability-II (The Rules of Probability & Counting Rules). QSCI 381 – Lecture 8 (Larson and Farber, Sects 3.3+3.4). Independent Events-I.

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Probability-II (The Rules of Probability & Counting Rules)

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  1. Probability-II(The Rules of Probability & Counting Rules) QSCI 381 – Lecture 8 (Larson and Farber, Sects 3.3+3.4)

  2. 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: Independent Note: Knowing that B has occurred doesn’t impact the probability of whether A will occur if A and B are independent.

  3. 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.

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

  5. 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).

  6. Mutually Exclusive Events Mutually exclusive • 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?

  7. The Additive Rule • The probability that events AorB will occur is: • If events A and B are mutually exclusive, then:

  8. Using the Rules of Probability(Example) • 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?

  9. Review of Concepts and Formulae

  10. 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.

  11. 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 1,2 ..10. 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….

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