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

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.

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

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

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.

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

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.

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)

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

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. 

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

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.

Classwork: CW 10/28, 1-8

Homework – None