Chapter 10: Introducing Probability STAT 1450

Connecting Chapter 10 to our Current Knowledge of Statistics Probability theory leads us from data collection (Chapter 8 & Chapter 9) to inference. The rules of probability will allow us to develop models so that we can generalize from our (properly collected) sample to our population of interest Introducing Probability

The Idea of Probability ▸ We can trust random samples and randomized comparative experiments because of chance behavior—chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run The Idea of Probability

Technology Tips—Generating Random Numbers ▸ TI–83/84 Apps  Prob Sim  (press any key)  Toss Coins. Click Toss. Select +1 (this is the Window key) (for your 2nd trial). Repeat selecting +1 (this is the Window key) 8 more times to obtain your first 10 trials. Now select +50 (the Trace key). Click on the Right Arrow key (to reveal the number of Tales). Click on the Right Arrow key again (to reveal the number of Heads). Repeat the last 3 steps to obtain the results for trials 61 – 110. Technology Tips—Generating Random Numbers

▸ JMP  Enter 1 into the top row of Column 1.  Right-click on Column 1. Select Formula.  Under the Functions (grouped) menu Select Random  Random Binomial.  In the fields provided, ENTER the: sample size (10 for the first time, then 50) and the probability (.50).  Click Apply. This returns the number of heads for the first 10 flips.  Repeat the process with Column 2. Use a sample size of 50 instead of 10. Technology Tips—Generating Random Numbers

Predicting Chance Behavior ▸ Example: Let’s observe the behavior of 10 coin flips The Idea of Probability # of Flips Expected Proportion of Heads Actual # of Heads Actual # of Tails Actual Proportion of Heads Complete the chart for 60 and 110 flips

Predicting Chance Behavior ▸ Example: Let’s observe the behavior of 10 coin flips The Idea of Probability # of Flips Expected Proportion of Heads Actual # of Heads Actual # of Tails Actual Proportion of Heads Complete the chart for 60 and 110 flips

Predicting Chance Behavior ▸ Example: Let’s observe the behavior of 10 coin flips The Idea of Probability # of Flips Expected Proportion of Heads Actual # of Heads Actual # of Tails Actual Proportion of Heads Complete the chart for 60 and 110 flips

Predicting Chance Behavior ▸ Example: Let’s observe the behavior of 10 coin flips The Idea of Probability # of Flips Expected Proportion of Heads Actual # of Heads Actual # of Tails Actual Proportion of Heads Complete the chart for 60 and 110 flips

Predicting Chance Behavior ▸ What should happen to the long run proportion of heads as we take more flips? ▸ Probability theory informs us that the sample proportion of heads should ‘converge’ to the population proportion (50%) The Idea of Probability

Key Terminology—Randomness and Probability ▸ We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. ▸ The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions Randomness and Probability

Randomness and Probability ▸ Technically, physically tossing a coin or rolling a die is predictable. If theoretically, the same force, angles, etc… are applied, the results should remain the same. But, usually persons apply differing amounts to these variables, thus resulting in what appears to be a random process. ▸ Using technology simulations (via graphing calculators or JMP) restore the randomness to the process Randomness and Probability

Randomness and Probability ▸ Technically, physically tossing a coin or rolling a die is predictable. If theoretically, the same force, angles, etc… are applied, the results should remain the same. But, usually persons apply differing amounts to these variables, thus resulting in what appears to be a random process. ▸ Using technology simulations (via graphing calculators or JMP) restore the randomness to the process Randomness and Probability

Poll ▸ Enter your actual proportion of heads after 60 trials _____________ ▸ As the number of coin flips increases, the proportion of heads should become closer to 50%. a)Trueb)False 10.2 Randomness and Probability

Poll ▸ Enter your actual proportion of heads after 60 trials _____________ ▸ As the number of coin flips increases, the proportion of heads should become closer to 50%. a)Trueb)False 10.2 Randomness and Probability

Key Terminology—Probability Models ▸ The sample space S of a random phenomenon is the set of all possible outcomes. ▸ An event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space. For an event A, the probability that A occurs is denoted P(A). ▸ A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space S and a way of assigning probabilities to events Probability Models

Example: Sales Effectiveness ▸ In a sales effectiveness seminar a group of sales representatives tried two approaches to selling a customer a new automobile: the aggressive approach and the passive approach Probability Models Result SaleNo SaleTotals ApproachAggressive Passive Totals

Example: Sales Effectiveness ▸ Sample space of Approach & Result:  {Aggressive-Sale, Aggressive – No Sale, Passive – Sale, Passive – No Sale} ▸ Event:  ▸ Probability model: 10.3 Probability Models Result P(Result)

Example: Sales Effectiveness ▸ Sample space of Approach & Result:  {Aggressive-Sale, Aggressive – No Sale, Passive – Sale, Passive – No Sale} ▸ Event:  Sale of automobile ▸ Probability model: 10.3 Probability Models Result P(Result)

Example: Sales Effectiveness ▸ Sample space of Approach & Result:  {Aggressive-Sale, Aggressive – No Sale, Passive – Sale, Passive – No Sale} ▸ Event:  Sale of automobile ▸ Probability model: 10.3 Probability Models ResultSale P(Result)0.585

Example: Sales Effectiveness ▸ Sample space of Approach & Result:  {Aggressive-Sale, Aggressive – No Sale, Passive – Sale, Passive – No Sale} ▸ Event:  Sale of automobile ▸ Probability model: 10.3 Probability Models ResultSaleNo Sale P(Result)

Proportions and Probabilities ▸ Beginning with Chapter 10, we will transition away from what proportion of persons have some characteristic to what is the probability that some event occurs Probability Models

Basic Rules of Probability ▸ The probability of each event assumes a number on the closed interval [0,1]. ▸ All outcomes of the sample space must total to 1. ▸ Addition Rule for disjoint events.  Two events are disjoint if they have no events in common.  In these cases P(A or B)= P(A) + P(B). ▸ Complement Rule  For any event A, P(A does not occur) = 1 – P(A) Probability Rules

Example: Sales Effectiveness ▸ We are familiar with these rules already. From the earlier example regarding result of sale, P(Aggressive) = 54% Probability Rules

Example: Sales Effectiveness ▸ We are familiar with these rules already. From the earlier example regarding result of sale, P(Aggressive) = 54%. Using the complement rule, P(not Aggressive) = =.46 = P(Passive) 10.4 Probability Rules

Finite and Discrete Probability Models Discrete probability models have countable outcomes. These outcomes either assume fixed values or are Natural numbers {0, 1, 2, …} Finite and Discrete Probability Models

Finite and Discrete Probability Models ▸ Example: Within the next 30 seconds, write down as many Michael Jackson songs as possible Finite and Discrete Probability Models

Continuous Probability Models ▸ A continuous probability model assigns probabilities as areas under a density curve. ▸ The area under the curve and above any range of values is the probability of an outcome in that range Continuous Probability Models

Continuous Probability Models Recall from Chapter 3: A density curve is the overall pattern of a distribution. The area under the curve for a given range of values along the x- axis is the proportion of the population that falls in that range. A density curve has total area 1 underneath it Continuous Probability Models

Density Curves b. What proportion of the time is the January hi temperature below 38 degrees? 38

Density Curves b. What proportion of the time is the January hi temperature below 38 degrees? Area= (base)(height) = (38-35)*(1/5) 38

Density Curves b. What proportion of the time is the January hi temperature below 38 degrees? Area= (base)(height) = (38-35)*(1/5) 38 =3*.20 P(X<38)=.60

Example: Length of a Michael Jackson Song 10.6 Continuous Probability Models

Random Variables ▸ A random variable is a variable whose value is a numerical outcome of a random phenomenon. ▸ The probability distribution of a random variable X tells us what values X can take and how to assign probabilities to those values. ▸ Random variables can be discrete or continuous.  Discrete random variables have a finite list of possible outcomes.  Continuous random variables can take on any value in an interval, with probabilities given as areas under a curve Random Variables

Poll Many dentists completed anatomy coursework as undergraduates. Two variables are of extreme interest to them– the number of adult teeth a patient has and the size (mm) of their incisors. Are both of these variables continuous random variables? Why or why not? 10.7 Random Variables

Poll Many dentists completed anatomy coursework as undergraduates. Two variables are of extreme interest to them– the number of adult teeth a patient has and the size (mm) of their incisors. Are both of these variables continuous random variables? Why or why not? No. the number of adult teeth is discrete. The size of their incisors is a continuous Random Variables

Poll Many dentists completed anatomy coursework as undergraduates. Two variables are of extreme interest to them– the number of adult teeth a patient has and the size (mm) of their incisors. Are both of these variables continuous random variables? Why or why not? No. the number of adult teeth is discrete. The size of their incisors is a continuous 1. Number of songs in your iPod? 2.The lengths of each song on your iPod? 10.7 Random Variables

Poll Many dentists completed anatomy coursework as undergraduates. Two variables are of extreme interest to them– the number of adult teeth a patient has and the size (mm) of their incisors. Are both of these variables continuous random variables? Why or why not? No. the number of adult teeth is discrete. The size of their incisors is a continuous 1. Number of songs in your iPod? Discrete 2.The lengths of each song on your iPod? 10.7 Random Variables

Poll Many dentists completed anatomy coursework as undergraduates. Two variables are of extreme interest to them– the number of adult teeth a patient has and the size (mm) of their incisors. Are both of these variables continuous random variables? Why or why not? No. the number of adult teeth is discrete. The size of their incisors is a continuous 1. Number of songs in your iPod? Discrete 2.The lengths of each song on your iPod? Continuous 10.7 Random Variables

Personal Probability A personal probability is a number between 0 and 1 that expresses someone’s judgment of an event’s likelihood. Example: I believe there is a 20% chance of precipitation tomorrow. This is based upon no relative frequency, no scientific evidence, just intuition Personal Probabilities

Five-Minute Summary ▸ List at least three concepts that had the most impact on your knowledge of probability. ______________________________________________