Binomial Distribution and application .pptx

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**Title: Understanding Binomial Distribution**

**Description:**

Dive into the fundamentals of Binomial Distribution with this comprehensive PowerPoint presentation. Perfect for students, educators, and professionals in statistics, mathematics, and data science, this presentation covers:

- **Introduction to Binomial Distribution**: What it is, its importance, and where it is used.
- **Key Concepts**: Understanding trials, successes, failures, and the binomial experiment.
- **Mathematical Foundation**: The binomial formula, probability mass function, and cumulative distribution function.
- **Properties and Characteristics**: Mean, variance, and standard deviation of a binomial distribution.
- **Real-World Applications**: Examples from various fields like finance, healthcare, and quality control.
- **Visualization**: Graphs and charts to illustrate binomial distribution concepts.
- **Problem Solving**: Step-by-step solutions to sample problems.

Enhance your knowledge and teaching with clear explanations, visual aids, and practical examples. Whether you are preparing for exams, teaching a class, or looking to apply binomial distribution in your work, this presentation is a valuable resource.

1. Binomial Distribution CIS 2003

2. The Binomial Distribution The Binomial distribution is a special type of discrete probability distribution that results from a probability experiment with a limited number of "trials", in which any trial has only two possible outcomes. For example, The probability of getting a “head” when tossing a coin is 0.5. If you toss 8 coins, what is the probability that you get 3 heads or less? Trials = 8 Outcomes = Head (0.5), Tail (0.5) The probability that a light bulb is “satisfactory” is 0.8. If you select 20 light bulbs, what is the probability that at least 15 of them are satisfactory? Trials = 20 Outcomes = Satisfactory (0.8), Not Satisfactory (0.2)

3. In a Binomial distribution, we need to know the following: n the number of trials p the probability of “success” of one trial q the probability of “failure” of one trial ( 1-p ) x the number of successes in the n trials n C x the number of ways that x successes can occur in n trials (most scientific calculators include a function called n C x ) The Binomial Distribution

4. Then, the probability of x successes in n trials is given by the formula A binomial distribution can then be created using the list of all P(x) values, from x = 0 to x = n . (Here p is the probability of success and q is the probability of failure.) The mean (expected value) of a binomial distribution is calculated by: The standard deviation of a binomial distribution is calculated by:   The Binomial Distribution

5. For example, an experiment is conducted in which 8 coins are tossed, and the number of heads that appear is recorded. In this case, n = 8 p = 0.5 q = 0.5 We can use Megastat- Excel to give us the probability distribution with the individual probabilities for each outcome ( x = 0 , x = 1 , … x = 8 ) The Binomial Distribution

6. Using Megastat for Binomial Distribution problems Instead of using the formula, one can use the Megastat Addin on Excel to analyse the data. All we need is the number of trials (n) and the probability (p) of the successful event. This also calculates the mean (µ), variance ( σ 2 ) and standard deviation ( σ ) of the distribution.

7. Using Megastat for Binomial Distribution A coin is tossed 3 times. What is the binomial distribution for number of tails that will result? For Megastat , we must know the number of trials (n) and the probability of success (p). 3 coins are tossed so n = 3. P(tossing a tail) = 0.5 To start Megastat : Open a new excel file. Open Megastat and click Enable Macros . Then click Add-ins to access Megastat on your excel sheet.

8. Click on arrow beside Megastat to find the function you need.

9. Click on Probability, then Discrete Probability Distributions to get the box below.

10. Enter number of trials (n = 3 since there are 3 coin tosses) and p of occurrence or success (p = 0.5 for tossing a tail). The click OK.

11. The output produced from Excel is shown here. The values of X, P(X) mean, variance and standard deviation for the distribution are thus obtained. Note that µ = 1.50 (expected value) σ 2 = 0.75 σ = 0.866 Look at the table to find the probability of tossing two tails. P(2 tails) = 0.375

12. What is the probability of tossing at most 1 tail? P(1) + P (0) = 0.37500 + 0.12500 = 0.50000 Or Cumulative probability from the table. P(1 or less) = 0.50000

13. What is the probability of tossing at least 1 tail? Either: P(1) + P(2) + P(3) = 0.87500 Or 1 – P(0) = 1 – 0.12500 = 0.87500

14. Another Example During a study by Health officials, it was found that 4 out of 25 restaurants in a city have unsatisfactory sanitary conditions. If a customer eats 6 times from restaurants in the city this month, how likely the customer will experience unsatisfactory conditions? Is this a binomial experiment? If it is binomial, create the binomial distribution.

15. Is it a binomial experiment? Fixed number of trials? Two outcomes only? Outcomes are independent? 4) Probability is constant for each trial? Yes. You will go out 6 times. Yes. Sanitary or unsanitary. Yes. The sanitary conditions at one restaurant do not affect those at another restaurant. The probability is always 4/25. It is a binomial experiment.

16. We need to know n and p. The number of trials (n) = 6. P(unsanitary conditions) = 4/25 = 0.16

17. What is the probability of eating at 2 restaurants that have unsanitary conditions? P = 0.19118 2) What is the probability of eating at more than 4 restaurants that have unsanitary conditions? p(5) + p(6) = 0.00053 + 0.00002 = 0.00055

18. What is the probability of eating in at least 1 restaurant that has unsanitary conditions? You can add up all of the probabilities of P(1) to P(6). 0.40148 + 0.19118 + 0.04855 + 0.00694 + 0.00053 + 0.00002 Or Take 1 – P(0) = 1 – 0.35130 = 0.6487

19. What is the probability of eating in at most 2 restaurants that have unsanitary conditions? P(2) + P(1) + P(0) = 0.019118 + 0.40148 + 0.35130 = 0.94396 or use cumulative probability from the chart. = p(2 or less) = 0.94396

20. What is the most likely number of times you will experience unsanitary conditions in the month? expected value or µ = 0.960 times How variable will the data be around that number? sd = 0.898

21. Example 3 The probability that a person shopping in Al Jimi mall will take advantage of a special promotion on ice cream is 0.30. Suppose 6 shoppers are selected at random. a) What is the probability that exactly 4 of these shoppers will take advantage of this promotion? n = 6 x = 4 Using formula p = 0.3 q = 0.7

22. b) What is the probability that at least 5 shoppers will take advantage of this promotion? n = 6 x = 0, 1, 2, 3, 4, 5 p = 0.3 f = 0.7 c) What is the expected value (mean) and standard deviation?

23. Using Megastat, You can then use basic Excel functions to calculate probability questions … For example, to calculate P(X ≥ 6), use =sum(C15:C19) = 0.96721

24. Example 4 The phone lines to the AAWC computer help desk are free only 60% of the time. Suppose that you plan to call the help desk 10 times today. Use Megastat to answer the following questions: a) What is the probability that the line will be free for exactly 3 of your calls? What is the probability that the line will be free for at least 1 of your calls? What is the mean and standard deviation for the number of times you can “expect” to get a free line?

25. Solution P(X=3) = 0.04247 P(X≥1) = 1 – (P(X=0)) = 1 – 0.0010 = 0.9999 E(x) = 6 s.d. = 1.549