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Binomial probability distributions ppt

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Tayab Ali
Tayab Ali

This presentation discusses binomial probability distributions through the following key points: - It defines basic terminology related to random experiments, events, and variables. The binomial distribution specifically describes discrete data from Bernoulli processes. - It outlines the notation and assumptions for binomial distributions, including that there are two possible outcomes for each trial (success/failure), a fixed number of trials, and constant probabilities of success/failure. - It presents three methods for calculating binomial probabilities: the binomial probability formula, table method, and using technology like Excel. - It discusses measures of central tendency and dispersion for binomial distributions and how the shape of the distribution depends on the number of trials and probability of success. - Real-world

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Slide 1Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. A Presentation On Binomial Probability Distributions By Tayab Ali (M/12/ME-11) ME- Industrial and Production Jorhat Engineering College
Slide 2 Outcome:- The end result of an experiment. Random experiment:- Experiments whose outcomes are not predictable. Random Event:- A random event is an outcome or set of outcomes of a random experiment that share a common attribute. Sample space:- The sample space is an exhaustive list of all the possible outcomes of an experiment, which is usually denoted by S. Basics and terminology
Slide 3 Basics and terminology (contd.) Mutually Exclusive Event. Random Variables.  Discrete Random Variable .  Continuous Random Variable. Binomial Distribution:- The Binomial Distribution describes discrete , not continuous, data, resulting from an experiment known as Bernoulli process.
Slide 4 Notation(parameters) for Binomial Distributions. S and F (success and failure) denote two possible categories of all outcomes. P(S) = p (p = probability of success) P(F) = 1 – p = q (q = probability of failure) n =denotes the number of fixed trials.
Slide 5 Notation(parameters) for Binomial Distributions( contd.) p =denotes the probability of success in one of the n trials. q =denotes the probability of failure in one of the n trials. P(x) =denotes the probability of getting exactly x successes among the n trials. • x = denotes a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive.
Slide 6 Assumptions for binomial distribution For each trial there are only two possible outcomes on each trial, S (success) & F (failure). The number of trials ‘ n’ is finite. For each trial, the two outcomes are mutually exclusive . P(S) = p is constant. P(F) = q = 1-p.  The trials are independent, the outcome of a trial is not affected by the outcome of any other trial. The probability of success, p, is constant from trial to trial.
Slide 7 Methods for Finding Probabilities  Method 1: Using the Binomial Probability Formula.
Slide 8 Method 1: Using the Binomial Probability Formula. For x = 0, 1, 2, . . ., n Where n = number of trials. x = number of successes among n trials. p = probability of success in any one trial. q = probability of failure in any one trial. (q = 1 – p).
Slide 9 Method 2: Table Method Part of A Table is shown below. With n = 12 and p = 0.80 in the binomial distribution, the probabilities of 4, 5, 6, and 7 successes are 0.001, 0.003, 0.016, and 0.053 respectively.
Slide 10 Method 3: Using Technology STATDISK, Minitab, Excel and the TI-83 Plus calculator can all be used to find binomial probabilities. STATDISK Minitab
Slide 11 Excel TI-83 Plus calculator
Slide 12 Measures of Central Tendency and dispersion for the Binomial Distribution. Mean, µ = n*p Std. Dev. s = Variance, s 2 =n*p*q Where n = number of fixed trials p = probability of success in one of the n trials q = probability of failure in one of the n trials
Slide 13 Shape of the Binomial Distribution The shape of the binomial distribution depends on the values of n and p. Fig.1.Binomial distributions for different values of p with n=10 •When p is small (0.2), the binomial distribution is skewed to the right. •When p= 0.5 , the binomial distribution is symmetrical. •When p is larger than 0.5, the distribution is skewed to the left.
Slide 14 Fig.2.Binomial distributions for different values of n with p=0.2 Fig. 2 illustrates the general shape of a family of binomial distributions with a constant p of 0.2 and n’s from 7 to 50. As n increases, the distributions becomes more symmetric.
Slide 15 Applications for binomial distributions Binomial distributions describe the possible number of times that a particular event will occur in a sequence of observations. They are used when we want to know about the occurrence of an event, not its magnitude. • In a clinical trial, a patient’s condition may improve or not. We study the number of patients who improved, not how much better they feel. •Is a person ambitious or not? The binomial distribution describes the number of ambitious persons, not how ambitious they are. •In quality control we assess the number of defective items in a lot of goods, irrespective of the type of defect. Examples
Slide 16 Areas of Application • Common uses of binomial distributions in business include quality control. Industrial engineers are interested in the proportion of defectives . • Also used extensively for medical (survive, die) • It is also used in military applications (hit, miss).
Slide 17 Thank You

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Binomial probability distributions ppt

  • 1. Slide 1Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. A Presentation On Binomial Probability Distributions By Tayab Ali (M/12/ME-11) ME- Industrial and Production Jorhat Engineering College
  • 2. Slide 2 Outcome:- The end result of an experiment. Random experiment:- Experiments whose outcomes are not predictable. Random Event:- A random event is an outcome or set of outcomes of a random experiment that share a common attribute. Sample space:- The sample space is an exhaustive list of all the possible outcomes of an experiment, which is usually denoted by S. Basics and terminology
  • 3. Slide 3 Basics and terminology (contd.) Mutually Exclusive Event. Random Variables.  Discrete Random Variable .  Continuous Random Variable. Binomial Distribution:- The Binomial Distribution describes discrete , not continuous, data, resulting from an experiment known as Bernoulli process.
  • 4. Slide 4 Notation(parameters) for Binomial Distributions. S and F (success and failure) denote two possible categories of all outcomes. P(S) = p (p = probability of success) P(F) = 1 – p = q (q = probability of failure) n =denotes the number of fixed trials.
  • 5. Slide 5 Notation(parameters) for Binomial Distributions( contd.) p =denotes the probability of success in one of the n trials. q =denotes the probability of failure in one of the n trials. P(x) =denotes the probability of getting exactly x successes among the n trials. • x = denotes a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive.
  • 6. Slide 6 Assumptions for binomial distribution For each trial there are only two possible outcomes on each trial, S (success) & F (failure). The number of trials ‘ n’ is finite. For each trial, the two outcomes are mutually exclusive . P(S) = p is constant. P(F) = q = 1-p.  The trials are independent, the outcome of a trial is not affected by the outcome of any other trial. The probability of success, p, is constant from trial to trial.
  • 7. Slide 7 Methods for Finding Probabilities  Method 1: Using the Binomial Probability Formula.
  • 8. Slide 8 Method 1: Using the Binomial Probability Formula. For x = 0, 1, 2, . . ., n Where n = number of trials. x = number of successes among n trials. p = probability of success in any one trial. q = probability of failure in any one trial. (q = 1 – p).
  • 9. Slide 9 Method 2: Table Method Part of A Table is shown below. With n = 12 and p = 0.80 in the binomial distribution, the probabilities of 4, 5, 6, and 7 successes are 0.001, 0.003, 0.016, and 0.053 respectively.
  • 10. Slide 10 Method 3: Using Technology STATDISK, Minitab, Excel and the TI-83 Plus calculator can all be used to find binomial probabilities. STATDISK Minitab
  • 11. Slide 11 Excel TI-83 Plus calculator
  • 12. Slide 12 Measures of Central Tendency and dispersion for the Binomial Distribution. Mean, µ = n*p Std. Dev. s = Variance, s 2 =n*p*q Where n = number of fixed trials p = probability of success in one of the n trials q = probability of failure in one of the n trials
  • 13. Slide 13 Shape of the Binomial Distribution The shape of the binomial distribution depends on the values of n and p. Fig.1.Binomial distributions for different values of p with n=10 •When p is small (0.2), the binomial distribution is skewed to the right. •When p= 0.5 , the binomial distribution is symmetrical. •When p is larger than 0.5, the distribution is skewed to the left.
  • 14. Slide 14 Fig.2.Binomial distributions for different values of n with p=0.2 Fig. 2 illustrates the general shape of a family of binomial distributions with a constant p of 0.2 and n’s from 7 to 50. As n increases, the distributions becomes more symmetric.
  • 15. Slide 15 Applications for binomial distributions Binomial distributions describe the possible number of times that a particular event will occur in a sequence of observations. They are used when we want to know about the occurrence of an event, not its magnitude. • In a clinical trial, a patient’s condition may improve or not. We study the number of patients who improved, not how much better they feel. •Is a person ambitious or not? The binomial distribution describes the number of ambitious persons, not how ambitious they are. •In quality control we assess the number of defective items in a lot of goods, irrespective of the type of defect. Examples
  • 16. Slide 16 Areas of Application • Common uses of binomial distributions in business include quality control. Industrial engineers are interested in the proportion of defectives . • Also used extensively for medical (survive, die) • It is also used in military applications (hit, miss).