Probability Class 28 1

Causation and Probability We are interested in finding the effect of Dr. Wong’s exploring teaching methodology on the statistics test score of a sample of TRMC students. Assume we could really isolate all other lurking variables and examine only the effect of Dr. Wong’s teaching methodology. If there is a gain in the test score, could we conclude that Dr. Wong’s teaching methodology works? 2

3 7.1 Random Circumstances Random circumstance is one in which the outcome is unpredictable. Outcome is NOT determined until we observe it. Alicia has a Bad Day VS A 40 year old man dies tomorrow VS Get a 2 out of rolling a die

Randomness and Probability What is the probability of getting a head when you toss a coin? Toss your coins ten times and record heads (H) or tails (T) on each toss. 4

Class Work What is the probability of getting a head when you toss a coin? Toss your coins ten times and record heads (H) or tails (T) on each toss. 5

Theoretical Probability VS Empirical Probability Theoretical Probability –the number of ways that the desired event can occur, divided by the total number of possible outcomes (sample space of known equally likely outcomes). Empirical Probability –is an "estimate" that the event will happen based on how often the event occurs after collecting data or running an experiment (in a large number of trials). 6

Randomness and probability A random phenomenon: individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions Probability: –The regular distribution of outcomes in terms of a proportion of times the outcome would occur in a large number of repetitions. –The value of probability is between 0 and 1 Probability describes the long-term regularity of random phenomenon. 7

8 Assigning Probabilities A probability is a value between 0 and 1 and is written either as a fraction or as a decimal fraction. A probability simply is a number between 0 and 1 that is assigned to a possible outcome of a random circumstance. For the complete set of distinct possible outcomes of a random circumstance, the total of the assigned probabilities must equal 1. The probability that an event does not occur is 1 minus the probability that the event does occur. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.

Classwork: Rolling two dice 1.What are the total possible outcomes? 2.What is the probability of each outcome? 3.What is the probability that the sum of the two dice is “5”?. 4.What is the probability that the sum of the two dice is not “5” 5.What is the probability that the sum of the two dice is “7” or “11”? 9

Interpretations of Probability The Relative Frequency Interpretation of Probability In situations that we can imagine repeating many times, we define the probability of a specific outcome as the proportion of times it would occur over the long run -- called the relative frequency of that particular outcome.

11 Example 7.1 Probability of Male versus Female Births Long-run relative frequency of males born in the United States is about Table provides simulation results: the proportion is far from over first few weeks but in long run settles down around

12 Determining the Relative Frequency Probability of an Outcome Method 1: Make an Assumption about the Physical World – Theoretical Probability Example 7.2 A 3 number Lottery – player winds if his or her three-digit number is chosen 1.What is the sample space? 2.What is the Theoretical Probability? 3.Does if mean you will win one time in every thousand plays?

13 Determining the Relative Frequency Probability of an Outcome Method 1: Make an Assumption about the Physical World – Theoretical Probability Example 7.3 Probability Alicia has to Answer a Question There are 50 student names in a bag. If names mixed well, can assume each student is equally likely to be selected. Probability Alicia will be selected to answer the first question is 1/50 or.02.

14 Determining the Relative Frequency Probability of an Outcome Method 2: Observe the Relative Frequency – Empirical Probability Example 7.4 The Probability of Lost Luggage “3.91 per thousand passengers on U.S. airline carriers will temporarily lose their luggage. ” Based on data collected over long run (a full year). Probability a randomly selected passenger on a U.S. carrier will temporarily lose luggage is 3.91/1000 = 1/256, or about

15 Proportions and Percentages as Probabilities Ways to express the relative frequency of lost luggage: The proportion of passengers who lose their luggage is 1/256 or about About 0.4% of passengers lose their luggage. The probability that a randomly selected passenger will lose his/her luggage is about

16 Estimating Probabilities from Observed Categorical Data Assuming data are representative, the probability of a particular outcome is estimated to be the relative frequency (proportion) with which that outcome was observed. Approximate margin of error for the estimated probability is

17 Example 7.5 Night-lights and Myopia Revisited Assuming these data are representative of a larger population, what is the approximate probability that someone from that population who sleeps with a nightlight in early childhood will develop some degree of myopia? Note: = 79 of the 232 nightlight users developed some degree of myopia. Estimated probability is 79/232 = Estimate based on sample of 232 with a margin of error of ~0.066.

18 The Personal Probability Interpretation Personal probability of an event = the degree to which a given individual believes the event will happen. Sometimes subjective probability used because the degree of belief may be different for each individual. For example: the hiring of a particular person Restrictions on personal probabilities: Must fall between 0 and 1 (or between 0 and 100%). Must be coherent.

Homework Assignment: Chapter 7 – Exercise 7.5, 7.7, 7.10 and 7.12 Reading: Chapter 7 – p