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Notes 8 - Notes 8 BI OMIAL DISTRIBUTIO Bernoulli Trials...

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Notes 8 BIglyph1197OMIAL DISTRIBUTIOglyph1197 Bernoulli Trials Bernoulli trials are trials that satisfy the following three conditions: head2right There are only two possible outcomes for each trial, called a “success” and “failure” head2right The probability of a success is the same for all trials head2right The trials are independent. A binomial situation arises when n independent trials (Bernoulli trials) of an experiment are performed. Examples : checkbld Tossing a coin 6 times where obtaining a head on a single toss is called the success and getting a tail is a failure. checkbld Rolling a die 10 times, where getting a 6 face up on a single throw is a success and not getting a 6 is a failure. Fixed number of trials Binomial Conditions : Trials are independent Only two possible outcomes at each trial Probability of success (p) is the same for all trials If X = the number of successes that occur in a fixed number of Bernoulli trials , then the appropriate probability model for X is the BIglyph1197OMIAL MODEL. Note : X is discrete with sample space S = {0,1,2,…n}. Therefore a Binomial Random Variable is used to describe the possible number of times that a particular event will occur in a sequence of trials that are independent and identical. We say X ~ Bin(n,p) glyph817ote: n and p are the parameters used to define the Binomial distribution. Examples : (i) A new drug is administered to a group of 20 patients. The probability of improvement is 0.7 for each patient. Let X = the number of patients who improved out of the 20 patients. X is a Binomial Random Variable. _______________________ (ii) A bag contains 2 red, 1 blue and 3 green balls. A person draws a ball with replacement on 10 occasions. Let G = the number of times a green ball is selected in the 10 draws. G ________ a Binomial Random Variable. _______________________ (iii) If the person decides to draw without replacement, is G still a Binomial r.v.?
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Notes 8 Calculating Probabilities for a Binomial Distribution A probability distribution usually has a distribution function associated with it that can be used to calculate probabilities for that particular type of distribution. The Binomial distribution is no exception. If X is a Binomial random variable with n trials, probability of success p (therefore probability of failure q=1-p ), then finding the probability that k successes occur requires two things: (1) The probability of any outcome in which there are k successes (and n – k failures): (p× p× …×p) × ( q× q×…×q ) = p k q n-k k successes n-k failures (2) The number of ways that these k successes can be chosen from n trials is: )! ( !
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