Math Tech IIII, Mar 6 The Binomial Distribution II Book Sections: 4.2 Essential Questions: How can I compute the probability of any event? What do I need to know about the binomial distribution? Standards: DA-5.6, DA-5.11, S.MD.1,.2,.3
What Makes a Binomial Experiment? A binomial experiment is a probability experiment that satisfies the following conditions: 1. Contains a fixed number of trials that are all independent. 2. All outcomes are categorized as successes or failures. 3. The probability of a success (p) is the same for each trial. 4. There is a computation for the probability of a specific number of successes.
Binomial Notation Binomial computations are known as probability by formula. The formula has a set of arguments that you must know and understand in application. Here is that notation: Symbol Description n The number of times a trial is repeated p The probability of success in a single trial q The probability of failure in a single trial (q = 1 p) x The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3,, n
The Inside Track A situation can be binomial if it consists of identical independent event trials. This results in the same probability at each trial Anything else is just smoke. The probability of exactly x in n trials is, by calculator: binomialpdf(n, p, x) where p is the probability of getting an x
The Next Level The probability of at most x successes means x This computation sums all discrete probability values up to and including x This is the purpose of the binomial cumulative function, binomialcdf on the calculator
Binomial Computation II Using binomial cdf (cumulative distribution function Use for the probability at most x successes in n trials Form is: binomialcdf(n, p, x) TI 83+: To get it, press [2 nd ] [DISTR] A [ALPHA MATH], enter arguments and enter. TI 84+: To get it, press [2 nd ] [DISTR] B [ALPHA APPS], enter arguments and enter.
Example You take a multiple choice quiz that has 10 questions. Each question has 4 multiple choice answers, of which 1 is correct. You complete the quiz by randomly selecting an answer to each question. The random variable x represents the number of correct answers. Compute the probability that you get at most 4 right.
Binomial Computation III Creating a binomial discrete probability distribution on the calculator: To construct a binomial distribution table, open STAT Editor 1) type in 0 to n in L1 2) Move cursor to top of L2 column (so L2 is hilighted) 3) Type in command binomialpdf(n, p, L1) and L2 gets the probabilities. 4) The distribution is now in L1 and L2.
Example You take a multiple choice quiz that has 10 questions. Each question has 4 multiple choice answers, of which 1 is correct. You complete the quiz by randomly selecting an answer to each question. The random variable x represents the number of correct answers. Produce a probability distribution for this situation.
Binomial Computations A binomialpdf computation or formula gives you the probability of exactly x successes in n trials. A binomialcdf (cumulative) computation gives you the probability of x or fewer (inclusive) [at most] successes in x trials. Fewer than x (or more than x) successes requires a sum or difference of more than one binomial probability computation. For this, you can: Use summation shorthand Add or subtract multiple binomial computations Add values from a binomial probability distribution table
Binomial Statistics Because of the nature of this distribution, binomial mean, variance, and standard deviation are almost trivial. Here are the formulas: μ = np σ 2 = npq σ = npq Mean Variance Standard deviation One other pearl of wisdom You could always compute mu and sigma using the 1-var stat L1, L2 computation on the calculator {providing you have the distribution in L1 and L2}
Example 1 R.H. Bruskin Associates Market Research found that 40% of Americans do not think having a college education is important to succeed in the business world. If a random sample of 5 Americans is selected, find these probabilities: A) Exactly 2 people agree with the statement B) At most, 3 people agree with the statement Produce a of this probability distribution and compute μ and σ
Example 2 An archer has a probability of hitting a target at 100 meters of 0.57. If he shoots 7 arrows, what is the probability that he hits the target: Exactly 3 times At most 4 times Fewer than 5 times
Example 3 In Pittsburgh, Pennsylvania, about 56% of the days in a year are cloudy. Find the mean, variance, and standard deviation of the number of cloudy days during the month of June.
Classwork: CW 3/6, 1-10 Homework None