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Binomial Probability

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Binomial Probability

To be considered to be a binomial experiment

Fixed number of trials denoted by n

n trials are independent and performed under identical conditions

Each trial has only two outcomes: success denoted by S and failure denoted by F

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)

The central problem is to determine the probability of r successes out of n trials. P(r) =

- 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

- There are several ways one can approach this problem
- Calculating it by hand
- Using a Binomial distribution table
- Using technology

- 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)

- 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) = Cn,x times px times qn-x
- P(4) = C5,4 0.254 0.755-4
- P(4) = 5 * 0.254 0.751
- P(4) = 5 * 0.0039 * 0.75 = 0.014625 ≈ 0.0146

By Using Binomial Probability table such as found at : 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

Excel = BIOMDIST(4,5,0.25,false) Ans: .0.014648438

TI 83 – 84

2nd 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.

- 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

- 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.