Math Tech IIII, Mar 13 The Binomial Distribution III Book Sections: 4.2 Essential Questions: What do I need to know about the binomial distribution? Standards: DA-5.6
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. The are computing 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 Formula When all else fails, have this up your sleeve and know how to use it. P( x) n C x p x q n x n = number of trials x = a desired number of successes p = probability of a success q = probability of a failure C is our old counting friend, the combination
Know the following: The List of Knows Recognize a binomial distribution, or when it applies in a probability problem Be able to compute binomial probability Know what is meant by a cumulative binomial distribution and when it applies Be able to create and use binomial probability distribution and be able to produce a binomial distribution histogram Know how to compute the mean (μ) and standard deviation (σ) of a binomial distribution
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) successes in x trials. The words at most x successes 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 C) Fewer than 3 people agree with the statement D) At least two people agree with the statement Produce a histogram 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 least 4 times Fewer than 5 times
Binomial Computation II Using binomial cdf (cumulative distribution function Use for the probability at most x successes in n trials (The previous formula) Form is: binomialcdf(n, p, x), you get probability of x successes in n trials. To get it, press [2 nd ] [DISTR] A [ALPHA MATH], enter arguments and enter.
Binomial Computation III Creating a binomial distribution and graph: 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) Go to stat plot and set up appropriate graph.
Classwork: Handout CW 3/13, 1-7 Homework None