What do you think "Binomial" involves?

Transcription

1 Learning Goals: * Define a binomial experiment (Bernoulli Trials). * Applying the binomial formula to solve problems. * Determine the expected value of a Binomial Distribution What do you think "Binomial" involves? A probability distribution based on the Binomial Theorem. Binomial Experiments are used in quality control processes where there are only two possible outcomes: pass or fail, use or don t use, good or bad, keep or discard, etc.

2 Binomial Experiments In order for an experiment to be deemed a binomial experiment, these properties must exist. 1. There must be two possible outcomes for each trial. These are normally referred to as success (p), and failure (q). 2. There are n identical trials which together form the binomial experiment. 3. The purpose of the experiment is to determine the # of successes that occur during the n trials. 4. The trials are independent of each other. 5. The probability of the outcomes remains the same from trial to trial. Binomial Distribution Any independent trial with 2 possible outcomes (success or failure) is called a Bernoulli trial. In any Binomial Experiment, the number of successes in n Bernoulli trials is a discrete random variable, X, called a binomial random variable. The resulting probability distribution is called a binomial distribution.

3 Formula Example 1: Recall : Coffee Experiment > You & friend flip coin and guess outcome to determine who pays for coffee each workday > What would expect to pay each week? Assume the coin toss is a Binomial Experiment Define success as a correct call of the coin toss p=0.5 The experiment is 5 trials (one for each week day) The discrete random variable, X, represents the number of wins in a week (5 tosses) Solution: Using the binomial probability distribution formula This leads to Which is equal to

4 Expected Value and Binomial Distributions Back to the example... We must again remember that our calculations here are theoretical in nature and may not reflect what could actually happen. Expected Value Notice where the expected value occurs on the graph of the distribution.

5 Example 2: Let s say that a cereal box could contain 1 out of 6 possible prizes. Determine the probability distribution for getting a specific prize in 4 boxes. Using a simulation, we will roll a die and let the number 2 be the specific prize we are looking for. a) Is the roll of the die a Bernoulli trial? Yes it is independent with 2 possible outcomes Success is a roll of 2 Failure is a roll of 2 b) Is this a Binomial experiment? (review 5 essential criteria) There must be two possible outcomes for each trial. These are normally referred to as success (p), and failure (q). There are n identical trials which together form the binomial experiment. The purpose of the experiment is to determine the # of successes that occur during the n trials. The trials are independent of each other. The probability of the outcomes remains the same from trial to trial. c) Identify the binomial random variable, X X is the number of 2 s occurring in 4 rolls. d) Find the probability that the 1st roll is a 2 and the others are not. e) Find the probability that one roll of 2 occurs in any one of the 4 rolls (boxes) and the other rolls are not a 2. Let a success be a roll of 2 We have n = 4 Bernoulli trials with p = 1/6 and q = 5/6 Let the random variable k = 1 f) Find the probability that two 2 s show up in the four rolls (boxes). Let a success be a roll of 2 We have n = 4 Bernoulli trials with p = 1/6 and q = 5/6 Let the random variable k = 2 g) Complete the theoretical probability distribution for all X (i.e. the number of 2 s showing in 4 rolls) h) Graph this as a Binomial Distribution Example 3: The Choco-Latie Candies company makes candy-coated chocolates, 40% of which are red. The production line mixes the candies randomly and packages 10 per box. 1. What is the probability that at least three candies in a given box are red? 2. What is the expected number of red candies in a box? Solution:

6 In groups you will present your answers to the class for the following questions: 1. Determine the probability, correct to four decimal places, that a die rolled six times in a row will produce the following. a) one 3 b) five 3s c) at least two 3s 2. A multiple choice quiz has 10 questions. Each question has four possible answers. Sam is certain that he knows the correct answer for questions 3, 5,and 8. If he guesses on the other questions, determine the probability that he passes the quiz. 3. In the dice game Yahtzee, a player has three tries at rolling some or all of a set of five dice. Each player is trying to achieve results such as three of a kind. If Cheryl rolls a pair of 2s on the first toss, and then rolls only the non 2s showing on the subsequent two tosses, find the probability that she gets a yahtzee. 4. What conditions must be satisfied in order for an experiment to be considered a binomial experiment? Describe a situation that meets these conditions.