5.1 Personal Probability

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1 5. Probability Value Page Personal Probability Although we think probability is something that is confined to math class, in the form of personal probability it is something we use to make decisions on a constant basis. We assign a probability to decisions we have to make and respond accordingly. There is debate over the definition of personal probability, often called decision theory. According to Suppes(1), there are only five personal probabilities. surely true more probable than not as probable as not (is there a better term for this one?) less probable than not surely false Personal Probability Below are examples where we might assign personal probabilities before making a decision: "Can I take a right on this red light?" "Can I ask this girl (or boy) out?" "Which checkout line do I join that will minimize my waiting?" "If I don't hand-in my homework on time will the teacher give me an extension?" "If I skip school, will I be caught?" "The weather on WRAL says 70% chance of rain. Do I go and play tennis?" "If I copy this student's work and hand-in this work as my own, will I get caught?" Even once we have assigned probabilities to these decisions as individuals, what factors play a role in our final decision? (1) From David Salsburg, "The Lady Tasting Tea" p.307, Holt 2001

5. Probability Value Page 2 5.2 Probability Value Defining 'Probability Value' Consider the experiment of rolling a single 6-sided die. 1.

2 5. Probability Value Page Probability Value Defining 'Probability Value' Consider the experiment of rolling a single 6-sided die. 1. What is the probability of rolling the die and getting a 1 a 2? 2. Sketch the distribution of the expected probabilities when you roll a 6-sided die. Don't forget to label the axes Shade the part of the distribution above which corresponds to question 1. What do you think 'probability value' means?

5. Probability Value Page 3 5.3 P-Value A 'probability value' is usually called a 'p-value'. Definition: p-value What are the highest and lowest values a p-value can have?

3 5. Probability Value Page P-Value A 'probability value' is usually called a 'p-value'. Definition: p-value What are the highest and lowest values a p-value can have? Terminology For our previous example, the probability of getting a one or a two on a sixsided die is: P(X 2)=⅓, where 'P' means probability and X is the outcome you are interested in. A p-value can be given as a fraction, decimal or percentage. Note: It is a important that all p-values are written with the above terminology. Not doing this would be akin to leaving off units of measurement (e.g. cm² or inches) giving the answer less context. What does the y-axis for our die rolling distribution look like? Why are we not keen to have 'bars' here? If you added up all the 'areas' of the bars, what total would you get and why?

4 5. Probability Value Page Finding P-Values Exercise - Rolling 2 Die and Finding P-Values In this exercise we will explore the distribution of the total of 2 dice and find p-values. 1. You need to complete 'frequency' column of the following JMP data file: To achieve this, you need to complete the following 'probability space'. A probability space shows the possible outcomes from an experiment (three totals have been completed for you. Die B Die A Now take the 'total frequencies' for all the possible outcomes and put them into the 'Frequency' column in JMP. As a check, what should your total of these frequencies be? In JMP, Analyze Distribution. 'Two Dice Total' goes under Y,Columns and 'Frequency' under Freq. To see the correct labeling for the y-axis, go to the Hotspot under 'Two Dice Total' and select Histogram Options Prob Axis. Use the JMP analysis to answer the following, given that X= total of rolling two dice: a. b. c. P (X = 7) = P (X < 4 ) = P (X 10) = A probability space is just a way to show all the possible outcomes from an event. Other types of probability spaces include a probability tree and an exhaustive list. Example: Finding the possible outcomes from flipping two coins: PROBABILITY TREE LIST TABLE HH HT HT TT

5 5. Probability Value Page Review - Binomial Theorem Before calculating binomial probability, review of the binomial theorem is needed. Remember this? The general formula for the binomial theorem is:, which expands to, where denotes the corresponding binomial coefficient. Quite simply, the binomial theorem is a quick way to expand the powers of a binomial:.. The pattern for the powers of and are easy to see, but where do the coefficients come from?

6 5. Probability Value Page Calculating the Binomial Coefficient We can use Pascal's Triangle to calculate the coefficients... But from Algebra II and/or Pre-Calculus, we learned to use: Read 'n choose k', where n is the power of the binomial and k is the term of the expansion QUICK CALCULATION: Go to and type in "6c2" Example: expand Review Exercise Expand the following: 1) 2) 3)

7 5. Probability Value Page Binomial Probability A Binomial Experiment has the following assumptions: It has a fixed number of trials There are only two possible outcomes (called success and failure) The probability of success and failure never change Each trial in the experiment is independent Exercise: Flipping a coin five times and recording the number of heads. Decide on the following: Number of Trials = Five Number of possible outcomes = Two Define Success = Getting a Head P (Success) = P (A Head) = 0.5 Define Failure = Getting a Tail P (Failure) = P (A Tail) = 0.5 Does the outcome of one coin flip influence the next? No, therefore the coin flips are INDEPENDENT. Exercise: You already know Binomial Probability 1. For the experiment of flipping three coins where success is getting a head, calculate: a. b. c. d. P (0 Heads) = P(1 Head) = P (2 Heads) = P (3 Heads) = How does Binomial Probability relate to the Binomial Theorem?

8 5. Probability Value Page Binomial Probability II Given = the number of trials = number of successes = probability of success = the probability of failure The Binomial Probability Distribution states that the probability of trials is successes in What does do in the above formula? Calculates the number of ways of getting successes out of What does do in the above formula? Calculates the probability of getting successes and failures Example: For the event of flipping four coins simultaneously and X=number of heads, find P(0 Heads), P(1 Head), P(2 Heads), P(3 Heads) and P(4 Heads) n = 4, r = 0, 1, 2, 3 or 4, p = r = 0.5 P (0 Heads) = P(1 Head) = P(2 Heads) = P(3 Heads) = P(4 Heads) = How can we check our results? The sum of all possible outcomes add up to one.

9 5. Probability Value Page Expected Number of Successes Exercise: Find the Expected Number If I toss a coin 10 times, how many heads would I expect to get? If I roll a 6-sided die 30 times, how many times would I expect to get a 1? Given = the number of trials = number of successes = probability of success = the probability of failure What is the formula for the expected number of successes for a binomial probability distribution? Use your results from the exercise to help you.

5. Probability Value Page 10 5.10 Binomial Notation If an experiment follows binomial probability distribution, then we can write: Note: the symbol ' ' is read 'has the distribution'.

4. Find the probability of 3 wet days in a week Find the probability of at least 5 wet days in a week. What is the expected number of wet days in a week?

10 5. Probability Value Page Binomial Notation If an experiment follows binomial probability distribution, then we can write: Note: the symbol ' ' is read 'has the distribution'. What type of numeric random variable (i.e. type of data) must binomial examples be? Example: Binomial Probability The number of days wet in a week follow a binomial model, (must be Britain) Find the probability of 3 wet days in a week Find the probability of at least 5 wet days in a week. What is the expected number of wet days in a week? Is there a problem using a binomial model for this example? 7 trials (i.e. 7 days), where the probability of it raining on a given day is 0.6 numeric ordinal Formula -> Discrete Probability -> Binomial Probability (n, p, r) (trials, probability of success,# successes)

11 5. Probability Value Page Example (continued) - Binomial Analyze -> Distribution # Wet Days Frequencies Level Total Count N Missing 0 8 Levels Prob Find the probability of 3 wet days in a week With JMP P(X = 3) = By hand P(X = 3) = 2. Find the probability of at least 5 wet days in a week. P(X 3. What is the expected number of wet days in a week? expected days wet in a week 4. Is there a problem using a binomial model for this example? If it is wet on Tuesday, there is probably a higher chance of rain on Wednesday. There is a lack of independence in the model.

12 5. Probability Value Page Exercise - Binomial Probability Exercise: Binomial Probability On average my train is late on 45 journeys out of 100. Next week I shall be making five train journeys. Let X denote the number of times my train will be late. a. State the assumption which must be made for X to be modeled as binomial probability? Find the probability that my train will be late on all given journeys b. 25% of people lose their calculator. Assuming a binomial model find, for a class of 15 pupils, the probability that: a. Exactly 3 lose their calculator b. Less than 3 lose their calculator c. At least one loses their calculator 20% of batsmen are bowled first ball. A team has 11 batsmen and X is the number bowled first ball (called a 'golden duck'). a. What sport is this??? b. State one assumption for X to be modeled a binomial distribution. Is that assumption reasonable? c. Assuming the binomial model, what is the probability that four of the team are bowled first ball?

5. Probability Value Page 13 5.13 Exercise - Zener cards Exercise - Zener Cards With this question, you will explore binomial distributions through the application of Zener cards.

13 5. Probability Value Page Exercise - Zener cards Exercise - Zener Cards With this question, you will explore binomial distributions through the application of Zener cards. I have a friend who I wish to check their psychic ability. I decide to use Zener cards test this. I randomly look at a card without my friend seeing it, and they try to read my mind and guess the shape on the card. This is done thirty times. a. Show the probability distribution of the possible outcomes (i.e. find P(X=0), P(X=1),..., P(X=30). Use JMP to show your results. b. How many would I Expect my friend to get correct? c. What outcomes might you want to want to consider my friend being psychic and why? If you don't know what Zener cards are, Who You Gonna Call? d. Optional Further Reading: Think that psychic ability is more science fiction than science fact? You might want to read this article from the New York Times (Jan 06, 2011): Journal's Paper on ESP Expected to Prompt Outrage

14 5. Probability Value Page Normal Probability Distribution We know that the Normal Distribution is a 'naturally' occurring phenomenon. It can be seen with height and weight distribution of populations and distribution of SAT scores. It is useful to be able to find out if a population measurement is 'Normal' as the Normal Distribution has many useful properties. Characteristics of Normal Distributions The following datafile contains the test scores from three classes. The scores from these classes are approximately Normal. To view the distributions of the scores, Analyze Distribution, putting all the variables under Y,Columns. Try Hotspot Stack to see them better For each Class, find out the proportion of scores that were within one standard deviation of the mean. Use the mean and standard deviation from your JMP output. (hint: use the quantiles) Compare the three proportions what do you notice? Repeat 1. and 2. to find the proportion of scores that are two standard deviations from the mean. Repeat 1. and 2. to find the proportion of scores that are three standard deviations from the mean. Use what you have learned from 1-4 and write three general rules for the spread (or dispersion) of data that is Normally Distributed.

5. Probability Value Page 15 5.15 Properties of a Normal Distribution A Normal Distribution follows a 68-95-99.

This is an approximate rule for the proportion of the population that is ± 1, 2 and 3 standard deviations from the

15 5. Probability Value Page Properties of a Normal Distribution A Normal Distribution follows a rule. This is an approximate rule for the proportion of the population that is ± 1, 2 and 3 standard deviations from the mean. Quantiles from the Rule Given that the mean µ is the 50th quantile for a Normal Distribution, use this and the distribution below to find the quantiles Dispersion µ - 3σ µ - 2σ µ - σ µ µ + σ µ + 2σ µ + 3σ Quantile

16 5. Probability Value Page Z-Score If X is a random variable having a Normal Distribution, we write where = variance (standard deviation squared). If we know how many standard deviations a data point is from the mean, we can use the properties of the Normal Distribution to calculate the quantile. where is data point of interest and is the corresponding quantile (or percentile) What does the red shaded area represent in context to what we know so far? (Note that 60 is the data point.) The shaded area represents the quantile or percentile of x=60. Remember, a quantile tells you the percentage of values that are below your data point. How do we calculate quantiles of data points? Using the formula above is tedious, so instead we calculate the z-score of a data point using the following, called the Z-Score: z x What does the Z-Score above calculate? (Hint: consider the role of the denominator) The Z-score calculates how far a data point is from the mean. It returns a value in standard deviations.

https://castatistics.wikispaces.com/standard+normal+table Example: In a test, a student scores an 85.

17 5. Probability Value Page Normal Distribution Table Once we have calculated a z-score, we use a Normal Distribution table to calculate the corresponding quantile. Example: In a test, a student scores an 85. The class has a mean of 80 with standard deviation of 4. What percentile is the student for the test? Sketch your problem so you can see what sort of answer you're aiming for: Z-Score = Therefore, we need to find what percentile 1.25 standard deviations from the mean is: (partial table)

18 Therefore, the student is the 89.4th percentile. 5. Probability Value Page 18

19 Normal Density Curve (concerns area under the curve) 5. Probability Value Page 19

5. Probability Value Page 20 5.18 Working with the Z-Score Example 2: In the same test, a student was 20th percentile. What score did get achieve?

20 5. Probability Value Page Working with the Z-Score Example 2: In the same test, a student was 20th percentile. What score did get achieve? Sketch your problem: Area = 20% Z-Score =, we need to find the z-score for 20%, rearranging this gives = 76.6 was the student's score. nearest value to 20% = 0.20 Exercise: SAT scores In North Carolina in 2009, the mean of the distribution of scores for the mathematics section for Seniors who took the SAT that year was 511 with a standard deviation of 105. Source: ) Presume the data to be Normally Distributed. Using this information solve the following and sketch your solution to each question. Use the full standard Normal table to help you: If to get into UNC Chapel Hill you needed a 620 in mathematics, what percentile would you have to be? (Note: 620 is made up) If a student scored 450, what percentile are they for mathematics SAT in NC?

21 5. Probability Value Page Speeding up Calculations Rather than use the table, we can use software or similar to calculate this for us. This also makes our results more accurate. With Wolfram Alpha ( ), to calculate: A p-value from a z-score: Our percentile is the area to the left, so use the first value A percentile to a z-score: We are concerned with the area to the left of the data point, so we need left-tailed value.

22 5. Probability Value Page Exercise - Normal Distribution Exercise - Normal Distribution 1. Find the shaded areas for the following Normal Distributions 2. Find the value of x in each of the following Normal Distributions

23 5. Probability Value Page Exercise - Applied Problems Exercise - Applied Problems Use a simple sketch to help illustrate you answers In an examination it is known that the distribution of marks is a. What proportion of marks will exceed 55? b. What proportion of marks will be less than 45? c. If an A Grade is awarded to the top 5% of marks, what mark must someone achieve in order to get this grade? d. The bottom 20% of marks are classed as Fails. What range of marks does this represent? A machine produces components to any required length specification with a standard deviation of 1.5 mm. If all lengths less than 90 mm are to be rejected, and if this rejection rate is to be 2.5%, to what value should the mean length be adjusted? (Assume the distribution of lengths to be Normal) Observation of a very large number of cars at a certain point on a highway establishes that the speeds are normally distributed. 97.5% of cars have speeds less than 130 km/hr, and 33% of cars have speeds less than 110 km/hr. Determine the mean speed and the standard deviation. The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. For each part that follows, sketch and shade an appropriate Normal distribution. Show your work. a. What percent of pregnancies last less than 240 days (that s about 8 months)? b. What percent of pregnancies last between 240 and 270 days (roughly between 8 months and 9 months)? c. How long do the longest 20% of pregnancies last?

24 5. Probability Value Page Light Bulb Answers Even once we have assigned probabilities to these decisions as individuals, what factors play a role in our final decision? Some of the possibilities could be morality, fear, confidence and obedience to State and Federal Law to name just a few. Definition: p-value A p-value is the probability of an event happening. What are the highest and lowest values a p-value can have? 0 is the lowest (i.e. impossible) and 1 the highest (i.e. certain) What does the y-axis for our die rolling distribution look like? It shows the expected probability (or frequency) If you added up all the 'areas' of the bars, what total would you get and why? You would get a total of 1, which would represent the total of all possible outcomes (being certain). The pattern for the powers of and are easy to see, but where do the coefficients come from? Pascal's Triangle or Combinatorics. How does Binomial Probability relate to the Binomial Theorem? For, is the probability of success, is the probability of failure and is the number of trials. What does do in the above formula? Calculates the number of ways of getting successes out of What does do in the above formula? Calculates the probability of getting successes and failures How can we check our results? The sum of all possible outcomes add up to one. What is the formula for the expected number of successes for a binomial probability distribution? Use your results from the exercise to help you. The expected number of successes = What type of numeric random variable (i.e. type of data) must binomial examples be? Discrete random variables as we have a exact number of success from an exact number of trials. What does the red shaded area represent in context to what we know so far? (Note that 60 is the data point.) The shaded area represents the quantile or percentile of x=60. Remember, a quantile tells you the percentage of values that are below your data point. What does the Z-Score above calculate? (Hint: consider the role of the denominator) The Z-score calculates how far a data point is from the mean. It returns a value in standard deviations.

The normal distribution is a theoretical model derived mathematically and not empirically.

Sociology 541 The Normal Distribution Probability and An Introduction to Inferential Statistics Normal Approximation The normal distribution is a theoretical model derived mathematically and not empirically.