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binomial distribution

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- 1. BINOMIAL PROBABILITY DISTRIBUTION
- 2. BASICS & TERMINOLOGY: > Outcome:- The end result of an experiment. > Random experiment predictable. > Random Event:- A Experiments whose outcomes are not 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
- 3. > Random Variables. □ Discrete Random Variable . □ Continuous Random Variable. > Bernoulli Trial: A trial having only two possible outcomes is called Bernoulli trials. example: □ success or failure □ head or tail
- 4. > Binomial Distribution:- The Binomial Distribution describes discrete , not continuous, data, resulting from an experiment known as Bernoulli process.
- 5. A binomial experiment is one that possesses the following properties: ► The experiment consists of n repeated trials; ► Each trial results in an outcome that may be classified as a success or a failure (hence the name, binomial); ► The probability of a success, denoted by p, remains constant from trial to trial and repeated trials are independent. ► The number of successes X in n trials of a binomial experiment is called a binomial random variable. ► The probability distribution of the random variable X is called a binomial distribution, and is given by the formula: P(X) = Cn xpxqn-x
- 6. EXAMPLE 1: ► Hospital records show that of patients suffering from a certain disease, 75% die of it. What is the probability that of 6 randomly selected patients, 4 will recover?
- 7. This is a binomial distribution because there are only 2 outcomes (the patient dies, or does not). Let X = number who recover. Here, // = 6 and ,v = 4. Letp = 0.25 (success - i.e. they live), <y = 0.75 (failure, i.e. they die). The probability that 4 will recover: PLY) = = C6 a( 0.25)4(0.75)2 = 15x 2.1973 xlO“3 0.0329595
- 8. EXAMPLE 3: In the old days, there was a probability of 0.8 of success in any attempt to make a telephone call. (This often depended on the importance of the person making the call, or the operator's curiosity!) Calculate the probability of having 7 successes in 10 attempts.
- 9. Probability of success p = 0.8, so q- 0.2. X=success in getting through. Probability of 7 successes in 10 attempts: Probability = P(X = 7) = Cj°(0.8)7 (0.2)10-7 = 0.20133
- 11. EXAMPLE 4: The ratio of boys to girls at birth in Singapore is quite high at 1.09:1. What proportion of Singapore families with exactly 6 children will have at least 3 boys? (Ignore the probability of multiple births.) [Interesting and disturbing trivia: In most countries the ratio of boys to girls is about 1.04:1, but in China it is 1.15:1.]
- 12. he probability of getting a boy is j QQ — 0. 521 5 et X = number of boys in the family ere, n = 6 , p = 0 5215, <7 = 1 - 0 52153 = 0 4785 - 3 P(X) - Cn xp*q- C|(0.5215)3(0.4785)3 - 0.31077 - 4 P(X) - C5(0.5215)4(0.4785)2 - 0.25402 - 5 P(X) - Cf(0.5215)5(0.4785)1 - 0. 1 1074 - 6 P(X) - C|(0.5215)6 (0.4785)° - 2. 0 1 1 5 x 10'2 o the probability of getting at least 3 boys is: Probability = P(X >3) - 0.31077+0.25402 + 0. 11074 + 2. Oil 5x 102 - 0.69565 NOTE: We could have calculated it like this: P{X > 3) - 1 - (PC*0) + P(*,) + P(*2))
- 13. EXAMPLE 5: A manufacturer of metal pistons finds that on the average, 12% of his pistons are rejected because they are either oversize or undersize. What is the probability that a batch of 10 pistons will contain (a) no more than 2 rejects? (b) (b) at least 2 rejects?
- 14. Let X = number of rejected pistons (In this case, "success” means rejection!) Here, n = 10, p = 0.12, q = 0.88. (a) No rejects x - 0 P(X) - C”p*q*~x - cj°(0. 12)0(0.88)'° One reject AT - 1 P(X) - C|°(0. 12)'(0.88)® - 0.37977 Two rejects x - 2 P(X) - Cj°(0. I2)2(0. 88)S - 0.23304 So the probability of getting no more than 2 rejects is: Probability = P(X < 2) = 0.278 5+ 0.3 797 7 4-0.233 04
- 15. ptoceed 3$ Mows. probabilityofatleast 2 reject* .t-(PW+PW) 3 -(0W+0.3WT) - 0.24H2
- 16. BINOMIAL FREQUENCY DISTRIBUTION ► If the binomial probability distribution is multiplied by the number of experiment N then the distribution is called binomial frequency distribution. f(X=x)= N P(X=x)=N v. n x qn x px
- 17. MEAN & VARIANCE: Mean, p = n*p Std. Dev. s = V ” * P Variance, s2 =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
- 18. 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. Example: > 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 ^^iaaQds,irrespective of the type of defect.