prob and stats, oct 28 the binomial distribution iii book sections: n/a essential questions: how can...
DESCRIPTION
Binomial Notation Binomial computations are known as probability by formula. The formula has a set of arguments that you must know and understand in application. Here is that notation: SymbolDescription n The number of times a trial is repeated p The probability of success in a single trial q The probability of failure in a single trial (q = 1 – p) x The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3, …, nTRANSCRIPT
![Page 1: Prob and Stats, Oct 28 The Binomial Distribution III Book Sections: N/A Essential Questions: How can I compute the probability of any event? How can I](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1af27f8b9ab05997f14e/html5/thumbnails/1.jpg)
Prob and Stats, Oct 28
The Binomial Distribution III
Book Sections: N/A
Essential Questions: How can I compute the probability of any event? How can I
compute a binomial probability distribution, easily?
Standards: PS.SPMD.1
![Page 2: Prob and Stats, Oct 28 The Binomial Distribution III Book Sections: N/A Essential Questions: How can I compute the probability of any event? How can I](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1af27f8b9ab05997f14e/html5/thumbnails/2.jpg)
What Makes a Binomial Experiment?• A binomial experiment is a probability experiment that
satisfies the following conditions:
1. Contains a fixed number of trials that are all independent.2. All outcomes are categorized as successes or failures.3. The probability of a success (p) is the same for each trial.4. There is a computation for the probability of a specific
number of successes.
![Page 3: Prob and Stats, Oct 28 The Binomial Distribution III Book Sections: N/A Essential Questions: How can I compute the probability of any event? How can I](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1af27f8b9ab05997f14e/html5/thumbnails/3.jpg)
Binomial Notation• Binomial computations are known as probability by
formula. The formula has a set of arguments that you must know and understand in application. Here is that notation:
Symbol Descriptionn The number of times a trial is repeatedp The probability of success in a single trialq The probability of failure in a single trial (q = 1 – p) x The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3, …, n
![Page 4: Prob and Stats, Oct 28 The Binomial Distribution III Book Sections: N/A Essential Questions: How can I compute the probability of any event? How can I](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1af27f8b9ab05997f14e/html5/thumbnails/4.jpg)
Binomial Computations• A binomialpdf computation or formula gives you the
probability of exactly x successes in n trials.• A binomialcdf (cumulative) computation gives you the
probability of x or fewer (inclusive) [at most] successes in x trials.
• Fewer than x (or more than x) successes requires a sum or difference of more than one binomial probability computation. For this, you can:Use summation shorthandAdd or subtract multiple binomial computationsAdd values from a binomial probability distribution table
![Page 5: Prob and Stats, Oct 28 The Binomial Distribution III Book Sections: N/A Essential Questions: How can I compute the probability of any event? How can I](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1af27f8b9ab05997f14e/html5/thumbnails/5.jpg)
Any Binomial Computation• The probability of any equality/inequality of x successes in
n trials. • Exactly x (x = ) binomialpdf(n, p, x)• At most x (x ≤ ) binomialcdf(n, p, x)Use these adjustments for any other inequality binomial
computation• Fewer than x (x <) binomialcdf(n, p, x -1)• At least x (x ≥) 1 – binomialcdf(n, p, x- 1)• More than x (x >) 1 – binomialcdf(n, p, x)
To use this sheet, always find n, p, and x in the basic problem, then adjust onto these computations.
![Page 6: Prob and Stats, Oct 28 The Binomial Distribution III Book Sections: N/A Essential Questions: How can I compute the probability of any event? How can I](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1af27f8b9ab05997f14e/html5/thumbnails/6.jpg)
Binomial Statistics• Because of the nature of this distribution, binomial mean,
variance, and standard deviation are almost trivial. Here are the formulas:
μ = npσ2 = npqσ =
One other pearl of wisdom – You could always compute mu and sigma using the 1-var stat L1, L2 computation on the calculator {providing you have the distribution in L1 and L2}
npq
Mean
Variance
Standard deviation
![Page 7: Prob and Stats, Oct 28 The Binomial Distribution III Book Sections: N/A Essential Questions: How can I compute the probability of any event? How can I](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1af27f8b9ab05997f14e/html5/thumbnails/7.jpg)
Binomial Computation III• Creating a binomial discrete probability distribution on
the calculator:To construct a binomial distribution table, open STAT
Editor 1) type in 0 to n in L12) Move cursor to top of L2 column (so L2 is hilighted)3) Type in command binomialpdf(n, p, L1) and L2 gets the
probabilities.4) The distribution is now in L1 and L2.
![Page 8: Prob and Stats, Oct 28 The Binomial Distribution III Book Sections: N/A Essential Questions: How can I compute the probability of any event? How can I](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1af27f8b9ab05997f14e/html5/thumbnails/8.jpg)
Example• You take a true-false quiz that has 10 questions. Each question has
2 choices of answer, of which 1 is correct. You complete the quiz by randomly selecting an answer to each question. The random variable x represents the number of correct answers. Produce a probability distribution for this situation.
x 0 1 2 3 4 5 6 7 8 9 10P(x)
![Page 9: Prob and Stats, Oct 28 The Binomial Distribution III Book Sections: N/A Essential Questions: How can I compute the probability of any event? How can I](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1af27f8b9ab05997f14e/html5/thumbnails/9.jpg)
Example• You take a true-false quiz that has 10 questions. Each question has
4 multiple choice answers, of which 1 is correct. You complete the quiz by randomly selecting an answer to each question. The random variable x represents the number of correct answers. Produce a probability distribution for this situation.
x 0 1 2 3 4 5 6 7 8 9 10P(x) .00098 .0098 .044 .117 .205 .246 .205 .117 .044 .0098 .00098
![Page 10: Prob and Stats, Oct 28 The Binomial Distribution III Book Sections: N/A Essential Questions: How can I compute the probability of any event? How can I](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1af27f8b9ab05997f14e/html5/thumbnails/10.jpg)
Example 2• An archer has a probability of hitting a target at 100 meters of 0.57.
If he shoots 5 arrows, create a probability distribution for the number of arrows that hit the target.
![Page 11: Prob and Stats, Oct 28 The Binomial Distribution III Book Sections: N/A Essential Questions: How can I compute the probability of any event? How can I](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1af27f8b9ab05997f14e/html5/thumbnails/11.jpg)
Example 3• An archer has a probability of hitting a target at 80 meters of 0.65.
If she shoots 9 arrows, what is the probability that she hits the target:
Between 5 and 7 times
![Page 12: Prob and Stats, Oct 28 The Binomial Distribution III Book Sections: N/A Essential Questions: How can I compute the probability of any event? How can I](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1af27f8b9ab05997f14e/html5/thumbnails/12.jpg)
What if?• Suppose that on a large campus, 2.5 percent of students
are foreign students. If 30 students are selected randomly, find the probability that the number of foreign students in the group will be between 2 and 8, inclusive.
![Page 13: Prob and Stats, Oct 28 The Binomial Distribution III Book Sections: N/A Essential Questions: How can I compute the probability of any event? How can I](https://reader035.vdocuments.us/reader035/viewer/2022070605/5a4d1af27f8b9ab05997f14e/html5/thumbnails/13.jpg)
Classwork: CW 10/28, 1-8
Homework – None