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To be considered to be a binomial experiment 1. Fixed number of trials denoted by n 2. n trials are independent and performed under identical conditions 3. Each trial has only two outcomes: success denoted by S and failure denoted by F 4. For each trial the probability of success is the same and denoted by p. The probability of failure is denote by q and p+q=1 (or q = 1 - p) 5. The central problem is to determine the probability of r successes out of n trials. P(r) =

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Understanding the Concept If the doctor tells you that the success rate for a given operation is 50%. That means that any given time the operation is performed there is a 50% chance of success. If the doctor performs 3 of these operations in a single day, the probability that all thee will be successful is 12.5%, it is also true that there is a 37.5% chance that 1 of the three will be successful. Where do these percentages come from? This is the discussion of the presentation

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Finding the P( x successes) There are several ways one can approach this problem Calculating it by hand Using a Binomial distribution table Using technology

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BASIC EXAMPLE Given 5 trials, with an historical probability of success on A SINGLE TRIAL of 25%. Of the 5 trials You can find the P(0 successes), P(1 success), P(2 successes), P(3 successes), P(4 successes), or P(5 successes). As an example, the following calculation will be for P(4 successes)

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BY HAND P( 4 successes) If n = 5 (number of trials) and p = 0.25, what is the probability of 4 successes let (x = 4)? P(4) = ? p + q = 1 so q = 1 – r = 1 – 0.25 = 0.75 Using the Formula on page 426 of text P(x) = C n,x times p x times q n-x P(4) = C 5,4 0.25 4 0.75 5-4 P(4) = 5 * 0.25 4 0.75 1 P(4) = 5 * 0.0039 * 0.75 = 0.014625 ≈ 0.0146

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By Table: P( 4 successes) By Using Binomial Probability table such as found at : http://www.uwsp.edu/math/hgonchig/Math_355/Tables/Binomial.pdf http://www.uwsp.edu/math/hgonchig/Math_355/Tables/Binomial.pdf -P( 4 successes) n=5, x=4, P(x) = 0.25 P(4) =.0146 http://www.uwsp.edu/math/hgonchig/Math_355/Tables/Binomial.pdf

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By Technology: P( 4 successes) Excel = BIOMDIST(4,5,0.25,false) Ans:.0.014648438 TI 83 – 84 2 nd DISTR choice 0 ENTER binompdf ( 5,0.25,4) ENTER Ans: 0.14648375 IF YOU USE ONE OF THESE TWO METHOD, EITHER ATTACHED THE EXCEL WORKBOOK, OR IF USING THE TI 83 OR 84 STATE THE FUNCTION AND ITS PARAMETERS.

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The Entire Probability Distribution Given 5 trials, with an historical probability of success on A SINGLE TRIAL of 25% P(of 0 success out of 5 trials) =.2373 P(of 1 success out of 5 trials) =.3955 P(of 2 successes out of 5 trials) =.2637 P(of 3 successes out of 5 trials) =.0879 P(of 4 successes out of 5 trials) =.0146 P(of 5 successes out of 5 trials) =.0010

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Summary You should be able to calculate a binomial probability by any of the three methods. Questions: post the slide number and your question to your individual forum.

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5-3 Binomial Probability Distributions

5-3 Binomial Probability Distributions

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