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binomial_distribution_real examples.pptx
- 1. Binomial Distribution With Real-LifeExamples Presented by Dr J Sinthiya Department of Mathematics Sri Ramakrishna College of Arts & Science Coimbatore
- 2. Introduction to BinomialDistribution • The binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states across a number of observations. • It is used when there are exactly two mutually exclusive outcomes of a trial, often referred to as 'success' and 'failure'.
- 3. Key Properties • Numberof trials (n) • Probability of success in a single trial (p) • Probability of failure (q = 1 - p) • Random variable X counts the number of successes in n trials • Follows the formula: P(X = k) = C(n, k) * (p^k) * (q^(n-k))
- 4. Real-Life Example 1:Coin Toss • Suppose a fair coin is tossed 10 times. • Probability of getting heads (success) in each toss: p = 0.5 • Let X be the number of heads obtained. • X follows a binomial distribution with parameters n = 10, p = 0.5.
- 5. Real-Life Example 2:Quality Control • A factory produces light bulbs with a 2% defect rate. • A random sample of 20 bulbs is tested. • Probability of finding exactly 1 defective bulb can be calculated using the binomial formula with n = 20, p = 0.02.
- 6. Real-Life Example 3:Exam Questions • A multiple-choice quiz has 5 questions, each with 4 possible answers. • Probability of guessing a question correctly: p = 0.25 • X = number of correct answers follows a binomial distribution with n = 5, p = 0.25.
- 7. Applications of BinomialDistribution • Quality control in manufacturing • Decision making in business • Genetics and biology studies • Risk analysis in finance • Sports performance analysis
- 8. Characteristics of BinomialDistribution • Fixed Number of Trials (n) The experiment consists of a fixed number of trials. Each trial is performed under the same conditions. • Two Possible Outcomes in Each Trial Each trial results in only two mutually exclusive outcomes: – Success (with probability p) – Failure (with probability q=1−p)
- 9. Characteristics of BinomialDistribution •Constant Probability of Success The probability p of success is the same for all trials. 𝑝 • Independent Trials The outcome of any trial does not affect the outcome of the other trials. • Discrete Random Variable The random variable counts the 𝑋 number of successes in trials. can take integer values from 𝑛 𝑋 0 to 𝑛.
- 10. Characteristics of BinomialDistribution •Mean and Variance • Mean (Expected Value): μ=np • Variance: npq • Standard Deviation: σ=npq • Shape of Distribution • Symmetrical when p=0.5 • Skewed to the right when p<0.5 • Skewed to the left when p>0.5
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