Intro to Probabiltiy theory notes for Elements Class.pptx

What is the probability that a woman picked at random

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What is the probability that a woman picked at random who used 2 or fewer drugs? What is the probability that a woman use 5 or more drugs?
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51 The Binomial Distribution Most widely encountered probability distribution. Derived from a process known as a Bernoulli trial. When a random process or experiment, called a trial can result only one of two mutually exclusive outcomes, such as dead or alive, sick or well, male or female, the trial is called a Bernoulli trial.
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52 The Binomial Distribution Before discussing the binomial distribution we need to look at the binomial theorem for a positive integer as a power. (a + b) 2 = (a + b)(a + b) = a 2 + ab + ba + b 2 = a 2 +2ab + b 2 (a + b) 3 = (a + b)(a 2 +2ab = b 2 ) = a 3 + 3a 2 b + 3ab 2 + b 3 This can be extended and the multipliers form Pascal’s triangle as below 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 1 0 1 0 5 1 1 6 1 5 2 0 1 5 6 1 Each number is the sum of the two numbers above it. (a+b) 0 (a+b) 1 (a+b) 2 (a+b) 3 …… Probability
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53 Characteristics of a binomial random variable variable The experiment consists of n identical trials There are only 2 possible outcomes on each trial. We will denote one outcome by S (for Success) and the other by F (for Failure). The probability of S remains the same from trial to trial. This probability will be denoted by p , and the probability of F will be denoted by q ( q = 1- p ). The trials are independent. The binomial random variable x is the number of S ’ in n trials .
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54 Binomial Table The calculation of probability using the probability mass function is tedious if the sample size is large. There are binomial probability tables are available where probabilities for different values of n, p and x have been tabulated.
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55 Binomial Parameters The binomial distribution has two parameters n and p which are sufficient to specify a binomial distribution. The mean of the binomial distribution is µ=np The variance of the binomial distribution σ = np(1-p) Strictly speaking the binomial distribution is applicable where the sampling is from an infinite population or finite population with replacement. But in practice the sampling is mostly without replacement. Hence question arises on the appropriateness of this distribution under these circumstances.
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56 The probability distribution: ( x = 0, 1, 2, ..., n ), Where, p = probability of a success on a single trial, q=1-p n = number of trials, x = number of successes in n trials combination of x from n = P x = n C x p x q n-x n C x = n! x!(n-x)!
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57 The Binomial Distribution In general the Binomial Expansion of when n is a positive integer is given by: (binomial theorem) n b) (a n n n n n n b b a n n n b a n n b a n a b a ... ! 3 ) 2 )( 1 ( ! 2 ) 1 ( ! 1 ) ( 3 3 2 2 1 Consider the expansion of (¾ + ¼) 5 here a = ¾, b = ¼ and n = 5 5 3 2 2 3 4 5 5 4 1 ...
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