Section 6.1 Discrete Random Variables Example: Probability Distribution, Spin the Spinners Sum of Numbers on Spinners Theoretical Probability 2 0.04 3 0.08 4 0.12 5 0.16 6 0.20 7 0.16 8 0.12 9 0.08 10 0.04 A random variable is a quantitative variable that represents a certain. of a probability experiment, and whose value depends on Two different kinds of random variables: Discrete random variable number of values for x. Continuous random variable on a continuous scale. A probability distribution Gives the probability for each value of the random variable X. Each probability must be. The sum of the probabilities must equal. P(2) = (means: probability of getting a sum exactly = 2) Page 1
Graphing a Discrete Probability Distribution Use the probability distribution to make a probability histogram, which is just a graph of the probability distribution. Values of the discrete random variable go on the. Probabilities of those values go on the. Interpreting the probability distribution/histogram: Describe the shape of the distribution: Compare the theoretical probability distribution to the empirical frequency histograms: Page 2
Disclaimer: We will be doing our discrete probability distribution graphs differently than is presented in the textbook (Sullivan 5 th edition). Example: From our book (Sullivan 5 th edition) Author s rationale in 5 th edition is that we want to emphasize that the data are discrete. From the previous edition of the book (Sullivan 4 th edition) Author s rationale in 4 th edition was to emphasize the relation between area and probability as a prelude to probabilities for continuous random variables. Page 3
Example: Coin flip experiment 1. Collect sample data from an experiment. Make a distribution and histogram from the experimental results. 2. Create a probability distribution based on the theoretical probabilities for the experiment. Make a probability distribution and histogram. Experiment is: Flip a coin 4 times. Count how many times you get out of 4 flips. Each group will repeat this experiment 10 times. x = random variable = P(x) = probability of that many out of 4 coin flips What are the possible values for the random variable x? Page 4
Experiment: data sheet Run your 10 trials, then summarize your data on the next page. Trial #1 Trial #2 Trial #3 Trial #4 Trial #5 Flip 1 Flip 1 Flip 1 Flip 1 Flip 1 Flip 2 Flip 2 Flip 2 Flip 2 Flip 2 Flip 3 Flip 3 Flip 3 Flip 3 Flip 3 Flip 4 Flip 4 Flip 4 Flip 4 Flip 4 Total H = Total H = Total H = Total H = Total H = Trial #6 Trial #7 Trial #8 Trial #9 Trial #10 Flip 1 Flip 1 Flip 1 Flip 1 Flip 1 Flip 2 Flip 2 Flip 2 Flip 2 Flip 2 Flip 3 Flip 3 Flip 3 Flip 3 Flip 3 Flip 4 Flip 4 Flip 4 Flip 4 Flip 4 Total H = Total H = Total H = Total H = Total H = Page 5
Summarize your results below in the first table then put them on the board (using tally marks). Each group will put a total of 10 tally marks on the board. Note: The frequency is the number of Trials in which you got x Heads, so for example, for x = 0, count up how many Total H = 0 results you have. Since you did 10 trials, the sum of the frequency column must equal 10. Your results: x (number of Heads in 4 flips) 0 Frequency 1 2 3 4 sum = 10 Class results: (don t fill this one in until we have ALL the data on the board) x (number of Heads in 4 flips) 0 1 2 3 4 Frequency sum = Relative Frequency Page 6
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Theoretical Model Start Toss #1 Toss #2 Toss #3 Toss #4 Outcome # heads H HHHH 4 H T HHHT 3 H H HHTH 3 T T HHTT 2 H H HTHH 3 H T HTHT 2 T H HTTH 2 T T HTTT 1 T H T H T H T H T H T H T H T x (number of Heads in 4 flips) 0 1 2 3 4 No. of ways it can occur total outcomes = Theoretical Probability P(x) Page 8
Mean Calculation for a Discrete Random Variable: Calculate the mean of the random variable: Multiply each value of x by its. Add up the column of. Interpret the mean of a discrete random variable: The mean outcome of the probability experiment if we repeated the experiment many times. Example: Mean number of Heads in Four Coin Flips Theoretical Distribution Experimental Distribution x P(x) x P(x) x P(x) x P(x) 0 0.0625 0 1 0.25 1 2 0.375 2 3 0.25 3 4 0.0625 4 sum = 1 Formula for mean of a discrete random variable: μ X = Round-off recommendation for and : decimal place than the values of the random variable. Note: always keep more decimal places for intermediate calculations Page 9
Standard Deviation Calculation for a Discrete Random Variable: Calculate the standard deviation of the random variable: Find the difference between value of the random variable and the. Square the difference. Multiply by the probability for that value. Example: Standard Deviation of Heads in Four Coin Flips x Standard Deviation of Theoretical Probability Distribution P(x) 0 0.0625 1 0.25 2 0.375 3 0.25 4 0.0625 x - ( = ) (x - ) 2 (x - ) 2 P(x) σ X 2 = variance = σ X = standard deviation = σ X 2 = Formula for standard deviation of a discrete random variable: σ X = Page 10
Example: Standard Deviation of Heads in Four Coin Flips x Standard Deviation of Experimental Probability Distribution P(x) x - ( = ) (x - ) 2 (x - ) 2 P(x) 0 1 2 3 4 sum = σ X 2 = variance = σ X = standard deviation = σ X 2 = Page 11
Section 6.2 The Binomial Probability Distribution A binomial probability distribution meets the following four requirements: 1. Procedure has a of trials. 2. Trials are. 3. Each individual trial has only possible categories of outcomes. 4. Probabilities of remain constant for each trial. The trials themselves are called, which are identical and independent repetitions of an experiment with two possible outcomes. Examples: Are these binomial procedures? 1. Roll a die 100 times and record the outcome 2. Roll a die 100 times and count how many 6s there are. 3. Spinning two spinners and adding the numbers (1 8) to find the sum. 4. Selecting 31 people from a very large population, and observing whether or not each of them is left-handed, when 10% of all people are known to be lefthanded. If a probability distribution is a binomial distribution, then there is a formula for calculating the probabilities for each value of the random variable. Three ways to use the formula to calculate probabilities for a binomial distribution: 1. 2. 3. Page 12
Notation for Binomial Probability Distribution Two outcomes are: = probability of success in one trial = probability of failure in one trial n = X = random variable representing total no. of in n trials P(x) = probability of successes out of n trials Success or failure : Somewhat of an arbitrary designation. Success doesn t necessarily mean something Success means the outcome, or what you are looking for. Key Point: corresponds to Both represent the same category of outcome, a Page 13
Binomial Probability Formula: P(x) = where, P(x) = probability of x successes in n trials n = number of trials x = number of p = probability of in one trial 1 p = probability of in one trial Example: Coin flip problem, probabilities of getting 0 4 heads in four flips x P(x) 0 0.0625 1 0.25 2 0.375 3 0.25 4 0.0625 1. Fixed number of trials? 2. Trials independent? 3. Outcomes in two categories? 4. Probabilities constant? Success = p = 1 p = n = x = (Example continued on next page) Page 14
Use binomial formula to calculate P(x) for values of x: P(x) = ncx p x (1 p) n-x x = 0: P(0) = x = 1: P(1) = x = 2: P(2) = There s no magic to the binomial formula it s based on: that we learned in Chapter 5. Example: From above, where x = 1 Which means: in 4 coin flips, have 1 H and 3 T Use the multiplication rule to calculate the probability of flipping a coin 4 times and getting: H, T, T, T P(H & T & T & T) = Why isn t this the same as what we calculated above for x = 1? So: P( x) x nx ncx p (1 p) First part of the formula tells us: Second part of the formula tells us: Page 15
Binomial Beans Sampling Experiment Every group has a cup with a total of 10 beans: 7 white beans 3 black beans The procedure that you re going to do is: Randomly select three beans (one at a time), and record their color. Notice that you are selecting WITH replacement, so put each bean BACK and shake up the cup before you select the next one. Each person in the group should perform the experiment for themselves, but then work together to answer the questions. Repeat the experiment TWO TIMES each. Define: s = success = choose a black bean 1. Record your results (every person do this individually): Trial # Color of Bean #1 Color of Bean #2 Color of Bean #3 # successes 1 2 2. The random variable x = number of successes, or number of black beans that you selected. On the board, each person put a tally mark next to the number of successes that you had. 3. Now you are going to calculate the probability for each of the values of the random variable x, using the Binomial Probability Formula. First, fill in the following: n = number of trials = p = probability of success in one trial = 1 p = probability of failure in one trial = Also, x = number of successes = 0, 1, 2 or 3 for this experiment Calculate probabilities: x P(x) = ncxp x (1 p) n-x 0 1 2 3 Page 16
Make a relative frequency histogram using the class experimental results: x Frequency Relative Frequency 0 1 2 3 Make a probability distribution using the theoretical probabilities: Page 17
4. How could you calculate these probabilities WITHOUT using the Binomial Probability Formula? Fill out the following table. See x = 0 for an example. Value of random variable x In other words, how many black beans were selected out of 3 trials 0 No black beans List the ways this could happen (B = black, W = white) Calculate the probability for each way using the Multiplication Rule: P = P(A)P(B)P(C) Add the probabilities from column 4 together this should be the same value that you calculated with the Binomial Formula. WWW (0.7)(0.7)(0.7) = 0.343 P(0) = 0.343 (3 ways) P(1) = 1 One black bean (3 ways) P(2) = 2 Two black beans (1 way) P(3) = 3 Three black beans Page 18
Table Look-up Method for Binomial Probabilities This method uses Table III in Appendix A Left-hand column is n, number of trials Next column is x, the number of successes Across the top are the different probabilities of success, p Example: Binomial Beans Example: Four Coin Flips n = n = p = p = x = x = Notice that the table look-up method can only be used for problems where the answers have been tabulated for the specific in the problem. Page 19
Using Technology to Calculate Binomial Probabilities Example: d. less than two. Using STATDISK: Analysis/Probability Distributions/Binomial Distribution Page 20
Mean and Standard Deviation for the Binomial Random Variable Any Discrete Probability Distribution Mean x P(x) Standard Deviation x X X P( x) X 2 Binomial Probability Distribution where, n = number of trials p = probability of 1 p = probability of Example: Food Safety (previous example) n = p = 1 p = Mean = μ X = Standard deviation = σ X = What μ X means: On average, if you ask 6 consumers in the US if they are confident that the food they buy is safe, about will say yes. Page 21
Shape of a Binomial Distribution Effect of p Trial size for all cases: n = Conclusion: for small trial sizes, the shape of the distribution is determined by. Shape of a Binomial Distribution Effect of n Conclusion: for a large trial size, the probability distribution becomes. Rule of thumb: if will be approximately bell-shaped., the probability distribution Page 22
Checking for Unusual Results in a Binomial Experiment 1. Using the Empirical Rule: Note: ONLY USE THIS if the distribution is approximately in other words if: maximum usual value minimum usual value Any values outside of these boundaries are considered to be Example: 4 coin flips Would it be unusual to get 0 Heads out of 4 flips? Page 23
Checking for Unusual Results in a Binomial Experiment (cont.) 2. Using probabilities: x successes out of a certain number of trials is unusually high if P(x or more) is. x successes out of a certain number of trials is unusually low if P(x or less) is. Note: USE THIS if the distribution is in other words if:. Key Point: Not just looking at. P(501) = P(501 or more) = Note: look at more or less based on which side of µ the value x is on. Probability Distribution for number of Heads out of 1000 coin flips 0.030 0.025 0.020 P(x) 0.015 Example: 0.010 0.005 0.000 400 406 412 Food Safety 418 424 430 436 442 448 454 460 466 472 478 484 Would it be unusual to get 1 or less people who are confident in their food? 490 x = number of Heads 496 502 508 514 520 526 532 538 544 550 556 562 568 574 580 586 592 598 Page 24
1. Each sample of air has a 10% probability of containing a particular rare molecule. Assume the samples are independent with regard to the presence of the rare molecule. Fifteen samples of air are taken at a time. Tip: Before calculating anything, start by identifying: S = success = what you are looking for in words = n = p = a. Find the probability that none of the fifteen samples contain the rare molecule. b. Find the probability that at least one of the fifteen samples contains the rare molecule. c. Find the probability that at most one of the fifteen samples contains the rare molecule. d. Find the mean and standard deviation of the number of air samples that contain the rare molecule. e. What will the shape of the probability distribution look like, and how do you know? f. Would it be unusual for four of the air samples out of fifteen to contain the rare molecule, and how did you determine this? Page 25
2. Clarinex-D is a medication whose purpose is to reduce the symptoms associated with a variety of allergies. In clinical trials of Clarinex-D, 5% of the patients in the study experienced insomnia as a side effect. (a) If 240 users of Clarinex-D are randomly selected, how many would we expect to experience insomnia as a side effect? (b) What is the probability that exactly 12 users out of the 240 will experience insomnia? (c) What will the shape of the probability distribution look like, and how do you know? (d) Would it be unusual to observe 20 patients experiencing insomnia as a side effect in 240 trials of the probability experiment, and how did you determine this? Page 26
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