Chap 5 probability, Lecture notes of Probability and Statistics

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A Survey of Probability

Concepts

Chapter 5

Learning Objectives

 (^) LO5-1 Define the terms probability , experiment , event , and outcome.  (^) LO5-2 Assign probabilities using a classical, empirical, or subjective approach.  (^) LO5-3 Calculate probabilities using the rules of addition.  (^) LO5-4 Calculate probabilities using the rules of multiplication.  (^) LO5-5 Compute probabilities using a contingency table.  (^) LO5-6 Calculate probabilities using Bayes’ theorem.  (^) LO5-7 Determine the number of outcomes using principles of counting.

Probability

LO5-1 Define the terms probability , experiment , event , and outcome. LO5-1 Define the terms probability , experiment , event , and outcome.  (^) Frequently expressed as a decimal, such as .70, .27, or .50.  (^) It may be given as a fraction such as 7/10, 27/100, or 1/2.  (^) The closer a probability is to 0, the more improbable it is the event will happen. The closer the probability is to 1, the more sure we are it will happen

Experiment, Outcome, and Event

 (^) An experiment is a process that leads to the occurrence of one and only one of several possible results.  (^) An outcome is the particular result of an experiment.  (^) An event is the collection/set of one or more outcomes of an experiment. LO5-

Experiment, Outcome, and Event Example

LO5-

Ways to Assign Probabilities

There are three ways to assign probabilities:

1. CLASSICAL PROBABILITY Based on the assumption that the outcomes of an experiment are equally likely.
2. EMPIRICAL PROBABILITY The probability of an event happening is the fraction of the time similar events happened in the past.
3. SUBJECTIVE PROBABILITY The likelihood (probability) of a particular event happening that is assigned by an individual based on whatever information is available. LO5-2 Assign probabilities using a classical, empirical, or subjective approach. LO5-2 Assign probabilities using a classical, empirical, or subjective approach.

Empirical Probability

The formula to determine an empirical probability is: Empirical approach to probability is based on what is called the Law of Large Numbers. LO5- The key to establishing probabilities empirically: a larger number of observations provides a more accurate estimate of the probability.

Empirical Probability - Example

On February 1, 2003, the Space Shuttle Columbia exploded. This was the second disaster in 123 space missions for NASA. On the basis of this information, what is the probability that a future mission is successfully completed?

Total number of flights

Number of successful flights

Probabilityof a successful flight

LO5-

Summarizing Probability

LO5-

Rules of Addition for Computing

Probabilities

 (^) Special Rule of Addition : If two events A and B are mutually exclusive, the probability of one or the other event occurring equals the sum of their probabilities. P(A or B) = P(A) + P(B)  (^) For three mutually exclusive events designated A, B, and C, the rule is written: LO5-3 Calculate probabilities using rules of addition. P(A or B or C) = P(A) + P(B) + P(C)

Special Rule of Addition - Example

Mutually Exclusive Events

A machine fills plastic bags with a mixture of beans, broccoli, and other vegetables. Most of the bags contain the correct weight, but because of the variation in the size of the beans and other vegetables, a package might be underweight or overweight. If a check of 4,000 packages filled in the past month revealed as follows: LO5-

Special Rule of Addition - Example

Mutually Exclusive Events

What is the probability that a particular package will be either underweight or overweight? P(A or C) = P(A) + P(C) = 0.025 + 0.075 = 0. The complement rule can also be used: Note that P(A or C) = P(~B), so P(~B) = 1 – P(B) = 1 - 0.900 = 0. LO5-

The Complement Rule - Example

An experiment has two mutually exclusive outcomes. Based on the rules of probability, the sum of the probabilities must be one. If the probability of the first outcome is 0.61, then logically, AND by the complement rule, the probability of the other outcome is (1.0 - 0.61) = 0.39. P (B) = 1 - P (~B) = 1 – 0. = 0. LO5-

Joint Probability

LO5-  (^) Sometimes, the outcomes of an experiment may not be mutually exclusive.  (^) For example, the survey of Florida Tourist Commission reveal that from 200 selecting tourist samples, there were 120 tourists went to Disney World and 100 tourists went to Busch Gardens. What is the probability that a person selected visited either Disney World or Busch Gardens? To solve an above problem, we use the rule to find a joint probability.