Probability and Probability Distributions Part III Probability and Probability Distributions

Chapter 5 Probability

Chapter 5 When we talk about probability, we are talking about a (mathematical) measure of how likely it is for some particular thing to happen Probability deals with chance behavior We study outcomes, or results of experiments Each time we conduct an experiment, we may get a different result Probability models the short-term behavior of experiments

Chapter 5 – Section 1 Rule – the probability of any event must be greater than or equal to 0 and less than or equal to 1 It does not make sense to say that there is a –30% chance of rain It does not make sense to say that there is a 140% chance of rain Note – probabilities can be written as decimals (0, 0.3, 1.0), or as percents (0%, 30%, 100%), or as fractions (3/10)

Chapter 5 – Section 1 If we do not know the probability of a certain event E, we can conduct a series of experiments to approximate it by This becomes a good approximation for P(E) if we have a large number of trials (the law of large numbers)

Chapter 5 – Section 1 Sometimes probabilities are difficult to calculate, but the experiment can be simulated on a computer If we simulate the experiment multiple times, then this is similar to the situation for the empirical method We can use

Chapter 5 – Section 1 Example We wish to determine what proportion of students at a certain school have type A blood We perform an experiment (a simple random sample!) with 100 students If 29 of those students have type A blood, then we would estimate that the proportion of students at this school with type A blood is 29%

Chapter 5 Descriptive statistics, describing and summarizing data, deals with data as it is Probability , modeling data, deals with data as it is predicted to be The combination of the two will let us do our inferential statistics techniques in Part IV

Discrete Probability Distributions Chapter 6 Discrete Probability Distributions

Overview These are probability distributions that are designed to model discrete variables Many of the discrete probability distributions model “counts”

Chapter 6 Sections Sections in Chapter 6 Discrete Random Variables The Binomial Probability Distribution 6.1 6.2

Discrete Random Variables Chapter 6 Section 1 Discrete Random Variables

Chapter 6 – Section 1 Learning objectives Distinguish between discrete and continuous random variables Identify discrete probability distributions Construct probability histograms Compute and interpret the mean of a discrete random variable Interpret the mean of a discrete random variable as an expected value Compute the variance and standard deviation of a discrete random variable 1 2 3 4 5 6

Chapter 6 – Section 1 Learning objectives Distinguish between discrete and continuous random variables Identify discrete probability distributions Construct probability histograms Compute and interpret the mean of a discrete random variable Interpret the mean of a discrete random variable as an expected value Compute the variance and standard deviation of a discrete random variable 1 2 3 4 5 6

Chapter 6 – Section 1 A random variable is a numeric measure of the outcome of a probability experiment Random variables reflect measurements that can change as the experiment is repeated Random variables are denoted with capital letters, typically using X (and Y and Z …) Values are usually written with lower case letters, typically using x (and y and z ...)

Chapter 6 – Section 1 Examples Tossing four coins and counting the number of heads The number could be 0, 1, 2, 3, or 4 The number could change when we toss another four coins Measuring the heights of students The heights could change from student to student Examples Tossing four coins and counting the number of heads The number could be 0, 1, 2, 3, or 4 The number could change when we toss another four coins

Chapter 6 – Section 1 A discrete random variable is a random variable that has either a finite or a countable number of values A finite number of values such as {0, 1, 2, 3, and 4} A countable number of values such as {1, 2, 3, …} Discrete random variables are designed to model discrete variables (see section 1.2) Discrete random variables are often “counts of …” A discrete random variable is a random variable that has either a finite or a countable number of values A finite number of values such as {0, 1, 2, 3, and 4} A countable number of values such as {1, 2, 3, …}

Chapter 6 – Section 1 An example of a discrete random variable The number of heads in tossing 3 coins (a finite number of possible values) There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads A finite number of possible values – a discrete random variable This fits our general concept that discrete random variables are often “counts of …” An example of a discrete random variable The number of heads in tossing 3 coins (a finite number of possible values) There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads A finite number of possible values – a discrete random variable An example of a discrete random variable The number of heads in tossing 3 coins (a finite number of possible values) There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads An example of a discrete random variable The number of heads in tossing 3 coins (a finite number of possible values)

Chapter 6 – Section 1 Other examples of discrete random variables The possible rolls when rolling a pair of dice A finite number of possible pairs, ranging from (1,1) to (6,6) The number of pages in statistics textbooks A countable number of possible values The number of visitors to the White House in a day Other examples of discrete random variables The possible rolls when rolling a pair of dice A finite number of possible pairs, ranging from (1,1) to (6,6) The number of pages in statistics textbooks A countable number of possible values Other examples of discrete random variables The possible rolls when rolling a pair of dice A finite number of possible pairs, ranging from (1,1) to (6,6) Other examples of discrete random variables

Chapter 6 – Section 1 A continuous random variable is a random variable that has an infinite, and more than countable, number of values The values are any number in an interval Continuous random variables are designed to model continuous variables (see section 1.1) Continuous random variables are often “measurements of …” A continuous random variable is a random variable that has an infinite, and more than countable, number of values The values are any number in an interval

Chapter 6 – Section 1 An example of a continuous random variable The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit The possible values (assuming that we can measure temperature to great accuracy) are in an interval The interval may be something like (–20,110) This fits our general concept that continuous random variables are often “measurements of …” An example of a continuous random variable The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit The possible values (assuming that we can measure temperature to great accuracy) are in an interval The interval may be something like (–20,110) An example of a continuous random variable The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit The possible values (assuming that we can measure temperature to great accuracy) are in an interval An example of a continuous random variable The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit

Chapter 6 – Section 1 Other examples of continuous random variables The height of a college student A value in an interval between 3 and 8 feet The length of a country and western song A value in an interval between 1 and 15 minutes The number of bytes of storage used on a 80 GB (80 billion bytes) hard drive Although this is discrete, it is more reasonable to model it as a continuous random variable between 0 and 80 GB Other examples of continuous random variables The height of a college student A value in an interval between 3 and 8 feet The length of a country and western song A value in an interval between 1 and 15 minutes Other examples of continuous random variables The height of a college student A value in an interval between 3 and 8 feet Other examples of continuous random variables

Chapter 6 – Section 1 The probability distribution of a discrete random variable X relates the values of X with their corresponding probabilities A distribution could be In the form of a table In the form of a graph In the form of a mathematical formula The probability distribution of a discrete random variable X relates the values of X with their corresponding probabilities

Chapter 6 – Section 1 If X is a discrete random variable and x is a possible value for X, then we write P(x) as the probability that X is equal to x Examples In tossing one coin, if X is the number of heads, then P(0) = 0.5 and P(1) = 0.5 In rolling one die, if X is the number rolled, then P(1) = 1/6 If X is a discrete random variable and x is a possible value for X, then we write P(x) as the probability that X is equal to x

Chapter 6 – Section 1 Properties of P(x) Since P(x) form a probability distribution, they must satisfy the rules of probability 0 ≤ P(x) ≤ 1 Σ P(x) = 1 In the second rule, the Σ sign means to add up the P(x)’s for all the possible x’s

Chapter 6 – Section 1 An example of a discrete probability distribution All of the P(x) values are positive and they add up to 1 x P(x) 1 .2 2 .6 5 .1 6

Chapter 6 – Section 1 An example that is not a probability distribution Two things are wrong P(5) is negative The P(x)’s do not add up to 1 An example that is not a probability distribution Two things are wrong P(5) is negative An example that is not a probability distribution Two things are wrong x P(x) 1 .2 2 .6 5 -.3 6 .1 x P(x) 1 .2 2 .6 5 -.3 6 .1 x P(x) 1 .2 2 .6 5 -.3 6 .1

Chapter 6 – Section 1 A probability histogram is a histogram where The horizontal axis corresponds to the possible values of X (i.e. the x’s) The vertical axis corresponds to the probabilities for those values (i.e. the P(x)’s) A probability histogram is very similar to a relative frequency histogram

Chapter 6 – Section 1 An example of a probability histogram The histogram is drawn so that the height of the bar is the probability of that value

Chapter 6 – Section 1 The mean of a probability distribution can be thought of in this way: There are various possible values of a discrete random variable The values that have the higher probabilities are the ones that occur more often The values that occur more often should have a larger role in calculating the mean The mean is the weighted average of the values, weighted by the probabilities The mean of a probability distribution can be thought of in this way: There are various possible values of a discrete random variable The values that have the higher probabilities are the ones that occur more often The values that occur more often should have a larger role in calculating the mean The mean of a probability distribution can be thought of in this way: There are various possible values of a discrete random variable The values that have the higher probabilities are the ones that occur more often The mean of a probability distribution can be thought of in this way: There are various possible values of a discrete random variable

Chapter 6 – Section 1 The mean of a discrete random variable is μX = Σ [ x • P(x) ] In this formula x are the possible values of X P(x) is the probability that x occurs Σ means to add up these terms for all the possible values x The mean of a discrete random variable is μX = Σ [ x • P(x) ]

Chapter 6 – Section 1 Example of a calculation for the mean Add: 0.2 + 1.2 + 0.5 + 0.6 = 2.5 The mean of this discrete random variable is 2.5 Example of a calculation for the mean Example of a calculation for the mean x P(x) x • P(x) 1 0.2 2 0.6 1.2 5 0.1 0.5 6 x P(x) 1 0.2 2 0.6 5 0.1 6 x P(x) x • P(x) 1 0.2 2 0.6 5 0.1 6 Multiply Multiply Multiply again Multiply again

Chapter 6 – Section 1 The calculation for this problem written out μX = Σ [ x • P(x) ] = [1• 0.2] + [2• 0.6] + [5• 0.1] + [6• 0.1] = 0.2 + 1.2 + 0.5 + 0.6 = 2.5 The mean of this discrete random variable is 2.5

Chapter 6 – Section 1 The mean can also be thought of this way (as in the Law of Large Numbers) If we repeat the experiment many times If we record the result each time If we calculate the mean of the results (this is just a mean of a group of numbers) Then this mean of the results gets closer and closer to the mean of the random variable The mean can also be thought of this way (as in the Law of Large Numbers) If we repeat the experiment many times If we record the result each time If we calculate the mean of the results (this is just a mean of a group of numbers) The mean can also be thought of this way (as in the Law of Large Numbers) If we repeat the experiment many times The mean can also be thought of this way (as in the Law of Large Numbers) The mean can also be thought of this way (as in the Law of Large Numbers) If we repeat the experiment many times If we record the result each time

Chapter 6 – Section 1 The expected value of a random variable is another term for its mean The term “expected value” illustrates the long term nature of the experiments – as we perform more and more experiments, the mean of the results of those experiments gets closer to the “expected value” of the random variable The expected value of a random variable is another term for its mean

Chapter 6 – Section 1 The variance of a discrete random variable is computed similarly as for the mean The mean is the weighted sum of the values μX = Σ [ x • P(x) ] The variance is the weighted sum of the squared differences from the mean σX2 = Σ [ (x – μX)2 • P(x) ] The standard deviation, as we’ve seen before, is the square root of the variance … σX = √ σX2 The variance of a discrete random variable is computed similarly as for the mean The mean is the weighted sum of the values μX = Σ [ x • P(x) ] The variance is the weighted sum of the squared differences from the mean σX2 = Σ [ (x – μX)2 • P(x) ] The variance of a discrete random variable is computed similarly as for the mean The mean is the weighted sum of the values μX = Σ [ x • P(x) ] The variance of a discrete random variable is computed similarly as for the mean

Chapter 6 – Section 1 The variance formula σX2 = Σ [ (x – μX)2 • P(x) ] can involve calculations with many decimals or fractions An equivalent formula is σX2 = [ Σ x2 • P(x) ] – μX2 This formula is often easier to compute

Chapter 6 – Section 1 For variables and samples (section 3.2), we had the concept of a population variance (for the entire population) and a sample variance (for a sample from that population) These probability distributions model the complete population These are population variance formulas There is no analogy for sample variance here

Chapter 6 – Section 1 The variance can be calculated by hand, but the calculation is very tedious Whenever possible, use technology (calculators, software programs, etc.) to calculate variances and standard deviations

Summary: Chapter 6 – Section 1 Discrete random variables are measures of outcomes that have discrete values Discrete random variables are specified by their Discrete probability distributions The mean of a discrete random variable can be interpreted as the long term average of repeated independent experiments The variance of a discrete random variable measures its dispersion from its mean

The Binomial Probability Distribution Chapter 6 Section 2 The Binomial Probability Distribution

Chapter 6 – Section 2 A binomial experiment has the following structure The first test is performed … the result is either a success or a failure The second test is performed … the result is either a success or a failure. This result is independent of the first and the chance of success is the same A third test is performed … the result is either a success or a failure. The result is independent of the first two and the chance of success is the same A binomial experiment has the following structure The first test is performed … the result is either a success or a failure The second test is performed … the result is either a success or a failure. This result is independent of the first and the chance of success is the same A binomial experiment has the following structure The first test is performed … the result is either a success or a failure A binomial experiment has the following structure

Chapter 6 – Section 2 Example Example Example Example Example A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit The card is then put back into the deck A second card is drawn from the deck with the same definition of success. The second card is put back into the deck We continue for 10 cards Example A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit The card is then put back into the deck A second card is drawn from the deck with the same definition of success. The second card is put back into the deck Example A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit The card is then put back into the deck Example A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit Example A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit The card is then put back into the deck A second card is drawn from the deck with the same definition of success.

Chapter 6 – Section 2 A binomial experiment is an experiment with the following characteristics The experiment is performed a fixed number of times, each time called a trial The trials are independent Each trial has two possible outcomes, usually called a success and a failure The probability of success is the same for every trial A binomial experiment is an experiment with the following characteristics The experiment is performed a fixed number of times, each time called a trial The trials are independent Each trial has two possible outcomes, usually called a success and a failure A binomial experiment is an experiment with the following characteristics The experiment is performed a fixed number of times, each time called a trial A binomial experiment is an experiment with the following characteristics A binomial experiment is an experiment with the following characteristics The experiment is performed a fixed number of times, each time called a trial The trials are independent

Chapter 6 – Section 2 Notation used for binomial distributions The number of trials is represented by n The probability of a success is represented by p The total number of successes in n trials is represented by X Because there cannot be a negative number of successes, and because there cannot be more than n successes (out of n attempts) 0 ≤ X ≤ n

Chapter 6 – Section 2 In our card drawing example Each trial is the experiment of drawing one card The experiment is performed 10 times, so n = 10 The trials are independent because the drawn card is put back into the deck Each trial has two possible outcomes, a “success” of drawing a heart and a “failure” of drawing anything else The probability of success is 0.25, the same for every trial, so p = 0.25 X, the number of successes, is between 0 and 10 In our card drawing example Each trial is the experiment of drawing one card The experiment is performed 10 times, so n = 10 The trials are independent because the drawn card is put back into the deck Each trial has two possible outcomes, a “success” of drawing a heart and a “failure” of drawing anything else The probability of success is 0.25, the same for every trial, so p = 0.25 In our card drawing example Each trial is the experiment of drawing one card The experiment is performed 10 times, so n = 10 The trials are independent because the drawn card is put back into the deck In our card drawing example Each trial is the experiment of drawing one card The experiment is performed 10 times, so n = 10

Chapter 6 – Section 2 We would like to calculate the probabilities of X, i.e. P(0), P(1), P(2), …, P(n) Do a simpler example first For n = 3 trials With p = .4 probability of success Calculate P(2), the probability of 2 successes

Chapter 6 – Section 2 For 3 trials, the possible ways of getting exactly 2 successes are S S F S F S F S S The probabilities for each (using the multiplication rule) are 0.4 • 0.4 • 0.6 = 0.096 0.4 • 0.6 • 0.4 = 0.096 0.6 • 0.4 • 0.4 = 0.096 For 3 trials, the possible ways of getting exactly 2 successes are S S F S F S F S S

Chapter 6 – Section 2 The total probability is But there is a pattern Each way had the same probability … the probability of 2 success (0.4 times 0.4) times the probability of 1 failure (0.6 times 0.6) The probability for each case is 0.42 • 0.61 The total probability is P(2) = 0.096 + 0.096 + 0.096 = 0.288 The total probability is P(2) = 0.096 + 0.096 + 0.096 = 0.288 But there is a pattern Each way had the same probability … the probability of 2 success (0.4 times 0.4) times the probability of 1 failure (0.6 times 0.6)

Chapter 6 – Section 2 There are 3 cases S S F could represent choosing a combination of 2 out of 3 … choosing the first and the second S F S could represent choosing a second combination of 2 out of 3 … choosing the first and the third F S S could represent choosing a third combination of 2 out of 3 These are the 3 = 3C2 ways to choose 2 out of 3 There are 3 cases S S F could represent choosing a combination of 2 out of 3 … choosing the first and the second S F S could represent choosing a second combination of 2 out of 3 … choosing the first and the third F S S could represent choosing a third combination of 2 out of 3

Chapter 6 – Section 2 Thus the total probability can also be written as P(2) = 3C2 • .42 • .61 In other words, the probability is The number of ways of choosing 2 out of 3, times The probability of 2 successes, times The probability of 1 failure Thus the total probability P(2) = .096 + .096 + .096 = .288 can also be written as P(2) = 3C2 • .42 • .61

Chapter 6 – Section 2 The general formula for the binomial probabilities is just this For P(x), the probability of x successes, the probability is The number of ways of choosing x out of n, times The probability of x successes, times The probability of n-x failures This formula is P(x) = nCx px (1 – p)n-x The general formula for the binomial probabilities is just this The general formula for the binomial probabilities is just this For P(x), the probability of x successes, the probability is The number of ways of choosing x out of n, times The probability of x successes, times The probability of n-x failures

Chapter 6 – Section 2 Example A student guesses at random on a multiple choice quiz There are n = 10 questions in total There are 5 choices per question so that the probability of success p = 1/5 = .2 What is the probability that the student gets 6 questions correct?

Chapter 6 – Section 2 Example continued This is a binomial experiment There are a finite number n = 10 of trials Each trial has two outcomes (a correct guess and an incorrect guess) The probability of success is independent from trial to trial (every one is a random guess) The probability of success p = .2 is the same for each trial

Chapter 6 – Section 2 Example continued The probability of 6 correct guesses is P(x) = nCx px (1 – p)n-x = 6C10 .26 .84 = 210 • .000064 • .4096 = .005505 This is less than a 1% chance In fact, the chance of getting 6 or more correct (i.e. a passing score) is also less than 1% Example continued The probability of 6 correct guesses is P(x) = nCx px (1 – p)n-x = 6C10 .26 .84 = 210 • .000064 • .4096 = .005505 This is less than a 1% chance Example continued The probability of 6 correct guesses is P(x) = nCx px (1 – p)n-x = 6C10 .26 .84 = 210 • .000064 • .4096 = .005505 Example continued The probability of 6 correct guesses is P(x) = nCx px (1 – p)n-x

Chapter 6 – Section 2 Binomial calculations can be difficult because of the large numbers (the nCx) times the small numbers (the px and (1-p)n-x) It is possible to use tables to look up these probabilities It is best to use a calculator routine or a software program to compute these probabilities

Chapter 6 – Section 2 We would like to find the mean of a binomial distribution Example There are 10 questions The probability of success is .20 on each one Then the expected number of successes would be 10 • .20 = 2 The general formula μX = n p

Chapter 6 – Section 2 We would like to find the standard deviation and variance of a binomial distribution This calculation is more difficult The standard deviation is σX = √ n p (1 – p) and the variance is σX2 = n p (1 – p)

Chapter 6 – Section 2 For our random guessing on a quiz problem x = 6 Therefore The mean is np = 10 • .2 = 2 The variance is np(1-p) = 10 • .2 • .8 = .16 The standard deviation is √.16 = .4 Remember the empirical rule? A passing grade of 6 is 10 standard deviations from the mean … For our random guessing on a quiz problem n = 10 p = .2 x = 6 For our random guessing on a quiz problem n = 10 p = .2 x = 6 Therefore The mean is np = 10 • .2 = 2 The variance is np(1-p) = 10 • .2 • .8 = .16 The standard deviation is √.16 = .4

Chapter 6 – Section 2 With the formula for the binomial probabilities P(x), we can construct histograms for the binomial distribution There are three different shapes for these histograms When p < .5, the histogram is skewed right When p = .5, the histogram is symmetric When p > .5, the histogram is skewed left With the formula for the binomial probabilities P(x), we can construct histograms for the binomial distribution

Chapter 6 – Section 2 For n = 10 and p = .2 (skewed right) Mean = 2 Standard deviation = .4

Chapter 6 – Section 2 For n = 10 and p = .5 (symmetric) Mean = 5 Standard deviation = .5

Chapter 6 – Section 2 For n = 10 and p = .8 (skewed left) Mean = 8 Standard deviation = .4

Chapter 6 – Section 2 Despite binomial distributions being skewed, the histograms appear more and more bell shaped as n gets larger This will be important!

Summary: Chapter 6 – Section 2 Binomial random variables model a series of independent trials, each of which can be a success or a failure, each of which has the same probability of success The binomial random variable has mean equal to np and variance equal to np(1-p)

Chapter 6 Summary Discrete probability distributions Random variables with discrete values Models counts Binomial distribution A sequence of success / failure trials Models the number of successes