Chapter 4 Probability Distributions 4-1 Random Variables 4-2 Binomial Probability Distributions 4-3 Mean, Variance, Standard Deviation for the Binomial Distribution 4-4 Other Discrete Probability Distributions

Overview probability distributions This chapter will deal with the construction of probability distributions by combining the methods of Chapter 2 with the those of Chapter 3. Probability Distributions will describe what will probably happen instead of what actually did happen. Emphasize the combination of the methods in Chapter 2 (descriptive statistics) with the methods in Chapter 3 (probability). Chapter 2 one would conduct an actual experiment and find and observe the mean, standard deviation, variance, etc. and construct a frequency table or histogram Chapter 3 finds the probability of each possible outcome Chapter 4 presents the possible outcomes along with the relative frequencies we expect

Combining Descriptive Statistics Methods and Probabilities to Form a Theoretical Model of Behavior

4-1 Random Variables

Definitions Random Variable Probability Distribution a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure Probability Distribution a graph, table, or formula that gives the probability for each value of the random variable

Probability Distribution Number of Girls Among Fourteen Newborn Babies x P(x) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0.000 0.001 0.006 0.022 0.061 0.122 0.183 0.209 Questions: Let X = # of girls P(X=3) = ? P(X ≤ 2) = ? P(X ≥ 1) = ? Point out the column that represents the random variable (x) and the column that represents the probabilities of each of those variables - P(x). The entire table is ‘the probability distribution’. Emphasize that the probability for any particular random variable is the probability of EXACTLY that number occurring. E.g.: P(9) = 0.122 is the ‘chance’ of getting EXACTLY 9 girls in 14 randomly selected newborns.

Probability Histogram Probability Histograms relate nicely to Relative Frequency Histograms of Chapter 2, but the vertical scale shows probabilities instead of relative frequencies based on actual sample results Observe that the probabilities of each random variable is also the same as the AREA of the rectangle representing the random variable. This fact will be important when we need to find probabilities of continuous random variables - Chapter 5.

Definitions Discrete random variable has either a finite number of values or countable number of values, where ‘countable’ refers to the fact that there might be infinitely many values, but they result from a counting process. Continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale with no gaps or interruptions. This chapter deals exclusively with discrete random variables - experiments where the data observed is a ‘countable’ value. Give examples. Following chapters will deal with continuous random variables.

Requirements for Probability Distribution  P(x) = 1 where x assumes all possible values 0  P(x)  1 for every value of x See #3 on hw

Mean, Variance and Standard Deviation of a Probability Distribution Formula (Mean) µ =  [x • P(x)] Formula (Variance) 2 =  [(x - µ)2 • P(x)] Formula (Standard Deviation)  = [(x - µ)2 • P(x)] In Chapter 2, we found the mean, standard deviation,variance, and shape of the distribution for actual observed experiments. The probability distribution and histogram can provide the same type information. These formulas will apply to ANY type of probability distribution as long as you have have all the P(x) values for the random variables in the distribution. In section 4 of this chapter, there will be special EASIER formulas for the special binomial distribution. The TI-83 and TI-83 Plus calculators can find the mean, standard deviation, and variance in the same way that one finds those values for a frequency table. With the TI-82, TI-81, and TI-85 calculators, one would have to multiply all decimal values in the P(x) column by the same factor so that there were no decimals and proceed as usual.

Roundoff Rule for µ, 2, and  Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round µ, 2, and  to one decimal place. If the rounding results in a value that looks as if the mean is ‘all’ or none’(when, in fact, this is not true), then leave as many decimal places as necessary to correctly reflect the true mean.

TI-83 Calculator Calculate Mean and Std. Dev from a Probability Distribution Press Stat Press “1” Edit Enter values of random variable (x) in L1 Enter probability P(x) in L2 Cursor over to CALC Choose the 1-Var stats option Enter 1-Var stats L1,L2

Using Excel See Probability Distribution Worksheet Examples: See Introduction to probability distributions handout Go to Excel (dice example – class assignment) Find missing probability, mean, SD and unusual values (test question see #6 on hw)

Unusual and Unlikely Values Unusual if greater than 2 standard deviations from the mean, that is x + 2 and x – 2s Unlikely if probability is very small, usually less than .05. This is consistent with the 2 idea associated with the empirical rule. #6f and 7c on HW If the rounding results in a value that looks as if the mean is ‘all’ or none’(when, in fact, this is not true), then leave as many decimal places as necessary to correctly reflect the true mean.

The average value of outcomes Definition Expected Value The average value of outcomes E =  [x • P(x)] Also called expectation or mathematical expectation Plays a very important role in decision theory

E = -$.07 E =  [x • P(x)] Event x P(x) x • P(x) Win Lose $5 - $5 Example: #9 on hw Event Win Lose x $5 - $5 P(x) 244/495 251/495 x • P(x) 2.465 - 2.535 E = -$.07

Binomial Random Variables 4-2 Binomial Random Variables

Binomial Random Variables Facts: Discrete (we can count the outcomes) Have to do with random variables having 2 outcomes. Examples: heads/tails, boy/girl, yes/no, defective/not defective, etc. A binomial distribution is the sum of several trials. Example: # of heads when a coin is tossed three times The word success is arbitrary and does not necessarily represent something GOOD. If you are trying to find the probability of deaths from hang-gliding, the ‘success’ probability is that of dying from hang-gliding. Remind students to carefully look at the probability (percentage or rate) provided and whether it matches the probability desired in the question. It might be necessary to determine the complement of the given probability to establish the ‘p’ value of the problem.

Notation for Binomial Probability Distributions n = fixed number of trials x = specific number of successes in n trials p = probability of success in one of n trials q = probability of failure in one of n trials (q = 1 - p ) P(x) = probability of getting exactly x successes among n trials The word success is arbitrary and does not necessarily represent something GOOD. If you are trying to find the probability of deaths from hang-gliding, the ‘success’ probability is that of dying from hang-gliding. Remind students to carefully look at the probability (percentage or rate) provided and whether it matches the probability desired in the question. It might be necessary to determine the complement of the given probability to establish the ‘p’ value of the problem.

Binomial Probability Formula Method 1 Binomial Probability Formula P(x) = • px • qn-x n ! (n - x )! x! P(x) = nCx • px • qn-x A minimal scientific calculator with factorial key will be required to complete the problems from this section. Most newer scientific calculators and all graphics calculators will have the ‘combinations’ key (nCr) which makes the computations much easier. Have students practice this formula with their calculator in class. Different calculators will have different key strokes to accomplish this formula. Many students stop after only computing the ‘counting’ factor of the formula. Remind that the formula has three factors and that the answer (a probability) should be a number between 0 and 1, inclusive. Remind students that for answers that are very small, their calculators might go into scientific notation. Look at the calculator notation very carefully. for calculators use nCr key, where r = x

Binomial Probability Formula P(x) = • px • qn-x (n - x )! x! Number of outcomes with exactly x successes among n trials Probability of x successes among n trials for any one particular order The remaining two factors of the formula will compute the probability of any one arrangement of successes and failures. This probability will be the same no matter what the arrangement is. The three factors multiplied together give the correct probability of ‘x’ successes.

Example: Toss a coin 3 times. Let x = number of heads and find P(2) = P(at least 2) This is a binomial experiment so you need to know 4 things p, q, n and x. p=.5 q=.5 n=3 a) x = 2 b) x = 0 then 1 then 2 On the test you will have to construct the entire probability distribution for tossing a coin “n” times and observing the number of heads. Should do this before the test.

Example: Find the probability of getting exactly 2 correct responses among 5 different requests from directory assistance. Assume in general, they are correct 80% of the time. This is a binomial experiment where: n = 5 x = 2 p = 0.80 q = 0.20 Using the binomial probability formula to solve: P(2) = 5C2 • 0.8 • 0.2 = 0.0512 This shows the formula using the ‘combination’ notation. 2 3

Method 2 Binomial Table Two tables are available on website

Example: Using Table for n = 5 and p = 0. 80, Example: Using Table for n = 5 and p = 0.80, find the following: a) The probability of exactly 2 successes b) The probability of at most 2 successes c) The probability of at least 1 success a) P(2) = 0.0512 b) P(at most 2) = P(0) or P(1) or P(2) = 0.0003 + 0.0064 + 0.0512 = 0.0579 c) P(at least 1) = 1 – P(0) = 1 – .0003 = .9997 Test Question

Using Technology Calculator function (TI-83) Method 3 Using Technology Calculator function (TI-83) See binomial distribution worksheet See also coin example worksheet

TI-83 Calculator Finding Binomial Probabilities (complete distribution) Press 2nd Distr Choose binopdf Enter binopdf(n,p) Press STO L2 (stores probabilities in column L2) Press Stat Choose Edit (to view probabilities) Optional: enter the values of the random variable in L1

TI-83 Calculator Finding Binomial Probabilities (individual value) Press 2nd Distr Choose binopdf Enter binopdf(n,p,x)

TI-83 Calculator Finding Binomial Probabilities (cummulative) Press 2nd Distr Choose binocdf Enter binocdf(n,p,x) This yields the sum of the probabilities from 0 to x. Example: Let n=6 and p=0.2 P(X<3) = binocdf(6,.2,3)

4.3 Mean, variance and standard deviation of a Binomial Probability Distribution Binomial probability distributions are important because they allow us to deal with circumstances in which the outcomes belong to TWO categories, such as pass/fall, acceptable/defective, etc.

For Any Discrete Probability Distribution the general formulas are: µ = [x • P(x)] 2=  [(x - µ) 2 • P(x) ] These formulas were introduced in section 4-1 These formulas will produce the mean, standard deviation, and variance for any probability distribution. The difficult part of these formulas is that one must have all the P(x) values for the random variables in the distribution.  =  [(x - µ) 2 • P(x)]

For Binomial Distributions: µ = n • p 2 = n • p • q = n • p • q The obvious advantage to the formulas for the mean, standard deviation, and variance of a binomial distribution is that you do not need the values in the distribution table. You need only the n, p, and q values. A common error students tend to make is to forget to take the square root of (n)(p)(q) to find the standard deviation, especially if they used L1 and L2 lists in a graphical calculator to find the standard deviation with non-binomial distributions. The square root is built into the programming of the calculator and students do not have to remember it. These formulas do require the student to remember to take the square root of the three factors.

= (20)(0.2)(0.8) = 1.8 answers (rounded) Example: Find the mean and standard deviation for students that guess answers on a multiple choice test with 5 answers and 20 questions. We previously discovered that this scenario could be considered a binomial experiment where: n = 20 p = 0.2 q = 0.8 Using the binomial distribution formulas: µ = (20)(0.2) = 4 correct answers = (20)(0.2)(0.8) = 1.8 answers (rounded) Test question

Reminder Maximum usual values = µ + 2  Minimum usual values = µ - 2  The Empirical Rule establishes these guidelines.

Example: Determine whether guessing 7 correct answers is unusual. For this binomial distribution, µ = 4 answers = 1.8 answers µ + 2  = 4 + 2(1.8) = 7.6 µ - 2  = 4 - 2(1.8) = .4 The usual number of correct answers would be from .4 to 7.6, so guessing 7 correct answers would not be an unusual result. Test question

4.4 Other Discrete Probability Distributions Poisson Geometric Hypergeometric Negative Binomial And more .

Poisson Distribution Definition a discrete probability distribution that applies to occurrences of some event over a specific interval. The random variable x is the number of occurrences of the event in an interval. The interval can be time, distance, area, volume, or some similar unit. Will be a question on the test for you to differentiate between a binomial and a poisson distribution

Definition Poisson Distribution P(x) = where e  2.71828 a discrete probability distribution that applies to occurrences of some event over a specific interval. Probability Formula Some discussion of the what the value of ‘e’ represents might be necessary. A minimal scientific calculator with factorial key is required for this formula. Most instructors will want students to use the calculator representation of ‘e’, rather than the rounded version of ‘e’, which might provide a very slightly different final answer. P(x) = where e  2.71828 µ x • e -µ x!

Example: Look at #1 Why is this a poisson distribution? µ = 5 We need to find various probabilities using Let’s find P(7) Look at the Poisson function in Excel µ x • e -µ P(x) = x!

Geometric Distribution Definition a discrete probability distribution of the number of trials needed to get one success. Will be a question on the test for you to differentiate between a binomial, poisson and a geometric distribution

Geometric Distribution Example: Roll a die 5 times. What is the probability of getting your first 2 on the 5th roll.

Negative Binomial Distribution Definition a discrete probability distribution of the number of trials needed to get a get a specified number of successes.

Negative Binomial Distribution Example a basketball player has a 70% chance of making a free throw, what is the probability of making his 3rd free throw on his 5th shot.

Hypergeometric Distribution Hypergeometric Experiment A sample of size n is randomly selected without replacement from a population of N items. . In the population, k items can be classified as successes, and N - k items can be classified as failures.

Hypergeometric Distribution Notation N: The number of items in the population. k: The number of items in the population that are classified as successes. n: The number of items in the sample. x: The number of items in the sample that are classified as successes. kCx: The number of combinations of k things, taken x at a time.

Hypergeometric Distribution Example Suppose we randomly select 5 cards without replacement from an ordinary deck of playing cards. What is the probability of getting exactly 2 red cards (i.e., hearts or diamonds)? P = [ kCx ] [ N-kCn-x ] / [ NCn ] = [ 26C2 ] [ 26C3 ] / [ 52C5 ] = [ 325 ] [ 2600 ] / [ 2,598,960 ] = 0.32513