Statistical Methods STAT 302 Chapter 3 Probability Copyright 2025 by Aburweis All rights reserved No part of this work may be reproduced in any form without written permission Chapter 3 Learning Objectives 1 Identify the sample space of a probability experiment 2 Calculate basic probabilities based on defined outcomes 3 Determine whether two events are mutually exclusive 4 Determine whether two events are independent 5 Apply the addition rule to calculate probabilities of compound events 6 Apply the multiplication rule to calculate joint probabilities 7 Calculate the expected value of a probability distribution 2 Why We Need Probability Studying probability is essential for the following reasons Probability is the cornerstone of modern science technology and decision making It provides a framework for quantifying uncertainty making informed predictions and understanding the likelihood of events Without probability our ability to navigate a world full of uncertainties would be severely limited Probability forms the foundational basis of statistical inference enabling researchers to draw conclusions and make predictions from data 3 Why We Need Probability Cont In both everyday life and professional settings we frequently encounter situations with uncertain outcomes For in genetics while the sex of an individual offspring cannot be predicted with certainty the long term proportions of male and female offspring follow predictable probabilities Similarly a physician might estimate that a patient has a 70 chance of surviving a specific surgery or another physician may express 95 certainty regarding a patient s diagnosis Probability theory provides a systematic and quantitative framework for measuring and expressing such uncertainties enabling more informed decision making instance 4 Randomness What Does Randomness Mean A phenomenon is considered random if individual outcomes or events are uncertain yet the outcomes exhibit a regular distribution over a large number of repetitions Random events are highly unpredictable in the short run but become predictable over the long run This is what keeps gamblers gambling but keeps casinos in business Caution The word random does not mean surprising unpredictable indiscriminate erratic without method or immeasurable These common dictionary definitions are not applicable in this course 5 Randomness Cont Determine whether the following scenarios illustrate random events and identify the possible outcomes 1 A COVID 19 test results randomly comes up positive 2 Your Professor randomly selects a student in the class from names written on slips of paper mixed into a bag 3 You bump into your best friend from third grade randomly in HEB Outcomes Outcomes Outcomes 6 Randomness Cont Determine whether the following scenarios illustrate random events and identify the possible outcomes 1 2 A COVID 19 test randomly comes up positive Not random Almost no false positives false negatives from not enough COVID in your nose Outcomes Positive Negative Your Professor randomly selects a student in the class from names written on slips of paper mixed into a bag Random the randomization method is using the bag of all names to choose one Outcomes Selecting Tyler selecting Jamie selecting Emma 3 You bump into your best Surprising doesn t mean random friend from third grade randomly in HEB Not random Outcomes Seeing your best friend not seeing your best friend 7 Probability Definitions Probability Experiment or Trial A process in which the outcome is determined by chance and cannot be predicted with certainty Not Haphazard Outcome Each individual result that is possible for a probability experiment Sample Space S The set that includes all possible outcomes of a given probability experiment Event A B A subset of the sample space It may consist of one outcome simple event or more outcomes compound event 8 Example 1 List the sample space and event Part 1 List the outcomes in the sample space for the following experiments Experiment 1 Toss a fair coin once Experiment 2 Toss a fair coin twice Experiment 3 Roll a fair six sided die one time Part 2 Define the event of observing one tail for experiment 1 observing one tail in experiment 2 and observing an even number in experiment 3 9 Example 1 List the sample space and event Solution Sample Space Experiment 1 Toss a fair coin once S H T Experiment 2 Toss a fair coin twice S HH HT TH TT Experiment 3 Roll a fair six sided die one time S 1 2 3 4 5 6 Event Experiment 1 toss a fair coin once S H T Event A observe one tail A T Experiment 2 Toss a fair coin twice S HH HT TH TT Event B observe one tail B HT TH Experiment 3 Roll a fair six sided die one time S 1 2 3 4 5 6 Event C observe an even number C 2 4 6 10 Your Turn 1 Consider an experiment in which a fair coin is tossed followed by a roll of a fair six sided die 1 List the outcomes in the sample space for the experiment 2 List the outcomes in the event tossing a tail then rolling an odd number 11 Your Turn 1 Solution 12 Probability Probability A number between 0 and 1 inclusive representing the likelihood that an event will occur Probabilities closer to 1 indicate that the event is more likely to occur while probabilities closer to 0 indicate that the event is less likely to occur Notation P A Read as Probability of A denotes the probability of event A For the complete set of distinct possible outcomes of a random experiment the sum of the assigned probabilities must equal 1 13 Assign Probability Probability can be assigned using several approaches The three primary methods are 1 Classical Probability Theoretical Probability 2 Empirical Statistical Probability 3 Subjective Probability 14 Assign Probability cont Classical probability Theoretical Probability When all outcomes of a probability experiment are equally likely the probability of an event is proportional to the number of outcomes that constitute the event The classical probability for an event is given by Number of outcomes in event Total number of outcomes in sample space Note This method assumes that all outcomes are equally likely 15 Example 2 Determining Classical Probabilities Consider rolling a fair six sided die Determine the probability of each event rounding to three decimal places 1 Event A Rolling a 3 2 Event B Rolling a 7 3 Event C Rolling a number less than 5 16 Example 2 Determining Classical Probabilities Solution The sample space S 1 2 3 4 5 6 With each outcome in the sample space is equally likely to
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