Binomial Probability Distributions Section 5-3 Binomial Probability Distributions

Binomial Probability Distribution A binomial probability distribution results from a procedure that meets all the following requirements: 1. The procedure has a fixed number of trials. 2. The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.) 3. Each trial must have all outcomes classified into two categories (commonly referred to as success and failure). 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. page 214 of Elementary Statistics, 10th Edition 4. The probability of a success remains the same in all trials.

Notation for Binomial Probability Distributions S and F (success and failure) denote the two possible categories of all outcomes; p and q will denote the probabilities of S and F, respectively, so P(S) = p (p = probability of success) 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. Page 214 of Elementary Statistics, 10th Edition. P(F) = 1 – p = q (q = probability of failure)

Notation (continued) n denotes the fixed number of trials. x denotes a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive. p denotes the probability of success in one of the n trials. q denotes the probability of failure in one of the n trials. Page 214 of Elementary Statistics, 10th Edition. P(x) denotes the probability of getting exactly x successes among the n trials.

Yes, all four requirements are met. Example 1: Determine whether or not the given procedure results in a binomial distribution. If it does not binomial, identify at least one requirement that is not satisfied. Treating 863 subjects with Lipitor (Atorvastatin) and recording whether there is a “yes” response when they are asked if they experienced a headache (based on data from Pfizer, Inc.) Yes, all four requirements are met. 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. page 214 of Elementary Statistics, 10th Edition

Example 2: Determine whether or not the given procedure results in a binomial distribution. If it does not binomial, identify at least one requirement that is not satisfied. Treating 152 couples with YSORT gender selection method developed by the Genetics & IVF Institute and recording the ages of the parents. No, there are more that two possible outcomes for the ages of the parents. 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. page 214 of Elementary Statistics, 10th Edition

Important Hints Be sure that x and p both refer to the same category being called a success. When sampling without replacement, consider events to be independent if n < 0.05N. Page 214-215 of Elementary Statistics, 10th Edition.

Methods for Finding Probabilities We will now discuss three methods for finding the probabilities corresponding to the random variable x in a binomial distribution. Page 216 of Elementary Statistics, 10th Edition.

Method 1: Using the Binomial Probability Formula for x = 0, 1, 2, . . ., n where n = number of trials x = number of successes among n trials p = probability of success in any one trial q = probability of failure in any one trial (q = 1 – p) page 216 of Elementary Statistics, 10th Edition

Binomial Probability Formula Number of outcomes with exactly x successes among n trials The 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 3: Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial. a) n = 12, x = 10, p = 3/4 b) n = 20, x = 4, p = 0.15

Method 2: Using Minitab Table Technology Minitab Table results can be used to find binomial probabilities. Minitab Page 218 of Elementary Statistics, 10th Edition.

a) Find the probability that at least 1 of the 5 donors Example 4: Refer to the Minitab display. (When blood donors were randomly selected, 45% of them had blood that is Group O (based on data from the Greater New York Blood Program).) The display shows the probabilities obtained by entering the values of n = 5 and p = 0.45. a) Find the probability that at least 1 of the 5 donors has Group O blood. b) If at least 1 Group O donor is needed, is it reasonable to expect that at least 1 will be obtained? Yes, it is reasonable to expect that at least one group O donors will be obtained since 0.950 > 0.05. x P(x) 0.050328 1 0.205889 2 0.336909 3 0.275653 4 0.112767 5 0.018453

Method 3: Using TI 83/84 Technology We will be doing binomial probabilities using the calculator:   *Keyword “exact” …find probability with binompdf(number of trials, probability of success, number of successes) *Keywords “less than or equal to” binomcdf(number of trials, probability of success, number of successes) **If it says “at least” or “greater than or equal to” use 1 – binomcdf with 1 less number for success What you are doing here is 1 minus what you DO NOT WANT and that will give you the probability that you do want.

Example 5: Use the TI 83/84 calculator to calculate the following probabilities. a) Calculate the probability of tossing a coin 20 times and getting exactly 9 heads. n = 20, p = 0.5, x = 9 binompdf(20, 0.5, 9) = 0.160     b) Calculate the probability of tossing a coin 20 times and getting less than 6 heads. n = 20, p = 0.5, x < 6 ≤ 5 binomcdf(20, 0.5, 5) = 0.021   c) Calculate the probability of tossing a coin 32 times and getting at least 14 heads. n = 32, p = 0.5, x ≤ 13 1 – binomcdf(32, 0.5, 13) = 0.811

Example 5 continued:   d) About 1% of people are allergic to bee stings. What is the probability that exactly 1 person in a class of 25 is allergic to bee stings?   n = 25, p = 0.01, x = 1 binompdf(25, 0.01, 1) = 0.196   e) Refer to (d), what is the probability that 4 or more of them are allergic to bee stings? n = 25, p = 0.01, x ≤ 3 1 – binomcdf(25, 0.01, 3) = 0.00011

Example 6: Use the TI 83/84 calculator to calculate the following probabilities. a) At a college, 53% of students receive financial aid. In a random group of 9 students, what is the probability that exactly 5 of them receive financial aid?   n = 9, p = 0.53, x = 5 binompdf(9, 0.53, 5) = 0.257      b) Now using (a), what is the probability that fewer than 3 students in the class receive financial aid? n = 9, p = 0.53, x ≤ 2 binomcdf(9, 0.53, 2) = 0.064

Example 7: The brand name of Mrs Example 7: The brand name of Mrs. Fields (cookies) has a 90% recognition rate (based on data from Franchise Advantage). If Mrs. Fields herself wants to verify that rate by beginning with a small sample of 10 randomly selected consumers, find the probability that exactly 9 of the 10 consumers recognize her brand name. Also find the probability that the number who recognize her brand name is not 9. n = 10, p = 0.90, x = 9 binompdf(10, 0.90, 9) = 0.387 1 – P(it is 9) = 1 – 0.387 = 0.613