Math Tech IIII, Apr 25

Transcription

1 Math Tech IIII, Apr 25 The Binomial Distribution I Book Sections: 4.2 Essential Questions: How can I compute the probability of any event? What is the binomial distribution and how can I use it? Standards: PS.SPMD.1

2 Probability Probability is, was, and will be a number between 0 and 1 (inclusive).

3 Simple Examples I roll a die 10 times. What is the probably of getting exactly two 4 s? A travel agency maintains a list of clients who like scuba diving trips. Sixty seven percent of the people on the list are men. They randomly select and call 10 clients from the list. What is the probability that they call exactly 6 men?

4 Binomial Experiments A binomial experiment is a random event where the outcome of each trial can be reduced to or categorized as two outcomes a success or failure. Some of our probability computations have been binomial experiments, but we did not call them by that label. In the binomial world, focus on the number of successes in any problem.

5 Definition of a Binomial Experiment A binomial experiment is a probability experiment that satisfies the following conditions: 1. The experiment is repeated for a fixed number of trials, where each trial is independent of the other trials. 2. There are only two possible outcome categories for each trial. The outcomes can be classified as a success (S) or as a failure (F). 3. The probability of a success (P(s)) is the same for each trial. 4. The random variable x counts the number of successful trials.

6 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 = P(s) The probability of success in a single trial q = P(f) 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

7 Identifying Binomial Experiments Is it or is not binomial? Apply the 4 conditions to a situation. If all fit, it is. If any one doesn t fit, its not. Binomial boils down to this The same thing is happening over and over, independently, and the probabilities of individual outcomes remain the same. Decide if the next three experiments are binomial. If they are, identify n, p, q, and list all possible values for the random variable, x.

8 Example 1 A certain surgical procedure has an 85% chance of success. A doctor performs the procedure on 8 patients. The random variable represents the number of successful surgeries.

9 Example 2 A jar contains five red marbles, nine blue marbles, and six green marbles. You randomly select three marbles from the jar, without replacement. The random variable x represents the number of red marbles.

10 A Cautionary Advisory If selecting from a small amount of objects means that selecting without replacement implies dependency. Remember our analysis in the probability unit where if you were selecting from a large group the conditional probability was not significantly affected by the previous selection? That was true with 52 objects. We will consider any large group (>60) to be independent in this unit, no matter what.

11 Example 3 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.

12 The Binomial Formula In a binomial experiment, the probability of exactly x successes in n-trials is: P( x) n C x p x q n x ( n n! x)! x! p x q n x

13 Binomial by Calculator The binomial distribution formula is pre-programmed into the calculator. With this nugget you can compute a: Binomial probability Cumulative binomial probabilities Create a binomial probability distribution And all of this right from the comfort of this classroom. Here s how

14 Binomial Computation Using binomial pdf (probability distribution function Use for the probability exactly x successes in n trials (The previous formula) Form is: binomialpdf(n, p, x), you get probability of x successes in n trials. TI 83+: To get it, press [2 nd ] [DISTR] 0, enter arguments and enter. TI 84+: To get it, press [2 nd ] [DISTR] A (Alpha Math), enter arguments and enter. TI-84+ Prompting Calculators newer versions of the 84+ prompt for the values. Trials = n, probability = p, successes = x.

15 Simple Examples I roll a die 10 times. What is the probably of getting exactly two 4 s? A travel agency maintains a list of clients who like scuba diving trips. Sixty seven percent of the people on the list are men. They randomly select and call 10 clients from the list. What is the probability that they call exactly 6 men?

16 A Test Drive A certain surgical procedure has an 85% chance of success. A doctor performs the procedure on 8 patients. The random variable represents the number of successful surgeries. Compute the probability of exactly 6 successes.

17 Another 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 3 right. Then compute the probability that you max the quiz.

18 The Starting Point In any binomial problem, find n, p, and x as a start.

19 Categorizing Outcomes In any binomial problem there can be more than two outcomes, but they are categorized in two ways. The test question problem is a good example. Each question has four answers. That is more than two. One answer is correct (that is a success), and three are wrong (those are categorized as failures).

20 Classwork: CW 4/25/16, 1-12 Homework None