Binomial and multinomial distribution

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1-Binomial distribution Binomial and multinomial distribution The binomial probability refers to the probability that a binomial experiment results in exactly "x" successes.

1 1-Binomial distribution Binomial and multinomial distribution The binomial probability refers to the probability that a binomial experiment results in exactly "x" successes. The probability of an event can be expressed as a binomial probability if the following conditions are satisfied. 1- There are a fixed number of trials (the number of trials is denoted by n). 2- The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials. 3- Each trial has only two possible outcomes; a success or a failure, and the probability of success, denoted by p, is the same on every trial. Consider the following example. A coin is tossed three times and if we want to count the number of times that the coin lands on heads, this will be a binomial experiment because: 1-The experiment consists of repeated trials (the coin is tossed 3 times). 2-Each trial can result in just two possible outcomes (Head or Tail). 3-The probability of success is constant (50%) on every trial. 4-The trials are independent; that is, getting Head on one trial does not affect whether we get Head on other trials. The formula for the binomial distribution From a binomial experiment consists of "n" trails and results in exactly "x" successes, the general formulate find the binomial probability is as follows: x: The number of successes that result from the binomial experiment. n: The number of trials in the binomial experiment. p: The probability of success on an individual trial. q: The probability of failure on an individual trial, where p and q are complementary (p + q = 1; q = 1 - p). Note that (0 x n) Example -1- Suppose a dice is tossed 5 times. What is the probability of getting exactly 2 fours? 1

167. Therefore, the binomial probability is: Example -2- An experiment consists of tossing a

2 This is a binomial experiment in which the number of trials is equal to 5, the number of successes is equal to 2, and the probability of success on a single trial is 1/6 or about Therefore, the binomial probability is: Example -2- An experiment consists of tossing a coin 8 times and counting the number of tails. Find the probability of seeing exactly 6 or 7 tails. 2

Example -3- If a student randomly guesses at 10 multiple - choice questions, find the probability that the student gets exactly 6 correct. Each question has five possible choices.

3 Example -3- If a student randomly guesses at 10 multiple - choice questions, find the probability that the student gets exactly 6 correct. Each question has five possible choices. 2-Multinomial distribution The multinomial distribution is similar to the binomial distribution but is more than two outcomes for each trial in the experiment. That is, the multinomial distribution is a general distribution, and the binomial is a special case of the multinomial distribution. The formula for the multinomial distribution Example -4- In a music store, a manager found that the probabilities that a person buys zero, one and two or more CDs are 0.3, 0.6 and 0.1 respectively. If six customers enter the store. Find the probability that one wanʼt buy any CDs, three will buy one CD, and two will buy two or more CDs? n = 6, x 1 = 1, x 2 = 3, x 3 = 2, p 1 = 0.3, p 2 = 0.6 and p 3 = 0.1 3

4 Example -5- In a large city, 50% of the people choose a movie, 30% choose dinner and a play, and 20% choose shopping. If a sample of five people is randomly selected. Find the probability that three are planning to go to a movie, one to a play and one to a shopping mall? n = 5, x 1 = 3, x 2 = 1, x 3 = 1, p 1 = 0.5, p 2 = 0.3 and p 3 = The Poisson distribution The Poisson distribution is a discrete distribution it is often used as a model for the number of events occurring over a periods of time. The Poisson distribution is useful when n is large and p is small. The major difference between Poisson and binomial distribution is that the Poisson dose not has a fixed number of trials. The formula for the Poisson distribution When a Poisson experiment is conducted, in which the average number of success within a given internal is λ, the poisson probability is: x: is the actual number of successes that result from the experiment and e is a constant (2.7183). P(x; λ): the poisson probability that exactly x successes occur in a poisson experiment, when the mean number of successes is λ. Example -6- If there are 200 typographical errors randomly distributed in a 500 page manuscript. Find the probability that a given page contain exactly three errors? 4

5 The mean number of errors (λ) is that there are 200 errors distributed over 500 pages, each page has an average of λ = 200/500 = 2/5 = 0.4 or 0.4 error per page. Since x=3, substituting into the formula yields: Thus, there is less a 1% probability that any given page will contain exactly three errors. Example -7- If approximately 2% of the people in a room of 200 people are left-handed, find the probability that exactly five people there are left-handed? λ = n.p λ = 200*0.02 = 4 Example -8- The average number of homes hold sold by a company is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow? This is a poisson distribution in which we know: λ = 2 x = 3, since we cannot to find the likelihood that 3 homes will be sold tomorrow. Substituting these values into the poisson formula yields: Thus, the probability of selling 3 homes tomorrow is 18%. 5

Chapter 3. Discrete Probability Distributions

Chapter 3. Discrete Probability Distributions Chapter 3 Discrete Probability Distributions 1 Chapter 3 Overview Introduction 3-1 The Binomial Distribution 3-2 Other Types of Distributions 2 Chapter 3 Objectives Find the exact probability for X successes