Math Tech IIII, Apr 30

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1 Math Tech IIII, Apr 30 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? Besides exactly, what other binomial probabilities are there? Standards: DA-5.6, S.MD.1,.2,.3

2 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.

3 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

4 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 a success.

5 Example Sixty two percent of New Yorkers say that they are satisfied with the current state of the city subway system. If 12 New Yorkers are randomly chosen and asked about the subway system today, what is the probability that exactly 5 of them will be satisfied with the system?

6 The Next Level The probability of at most x successes means x some value 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

7 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. The format is the same as before: binomialcdf(n, p, x) X is now a cutoff value.

8 What s the Difference? binomialpdf computes the value of a single whole number outcome and is used when you want an exact number of successes in binomial probability. binomialcdf computes the probability of every value up to and including a number of successes and adds them as a cumulative probability.

9 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.

10 In Any Binomial Computation Any time you are trying to accomplish a binomial computation, no matter the type, make this your first step: Identify the values of n, p, and x from the words of the problem. Usually: n is the big number x is the smaller number p is sometimes a percent (convert to a decimal) If no p value is found, use a theoretical value if it applies Sometimes a value will be implied

11 Any Binomial Computation The probability of any equality/inequality of x successes in n trials. Exactly x (x = ) binomialpdf(n, p, x) At most x (x ) binomialcdf(n, p, x) Use these adjustments for any other inequality binomial computation Fewer than x (x <) binomialcdf(n, p, x -1) At least x (x ) 1 binomialcdf(n, p, x- 1) More than x (x >) 1 binomialcdf(n, p, x)

12 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 4 people agree with the statement

13 Example 2 An archer has a probability of hitting a target at 100 meters of If he shoots 7 arrows, what is the probability that he hits the target: Exactly 3 times At least 4 times More than 4 times

14 Example 3 In Pittsburgh, Pennsylvania, about 56% of the days in a year are cloudy. What is the probability that there will be at least 15 cloudy days in Pittsburgh this February? At least x (x ) 1 binomialcdf(n, p, x- 1)

15 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.

16 Example You take a true-false quiz that has 6 questions. Each question has 2 possible 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.

17 Classwork: CW 4/30/15, 1-10 Homework None

Math Tech IIII, Mar 6

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: