Transcript 5 Minute Check, 26 Sep

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.
2.
3.
4.
Contains a fixed number of trials that are all independent.
All outcomes are categorized as successes or failures.
The probability of a success (p) is the same for each trial.
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
n
p
q
x
Description
The number of times a trial is repeated
The probability of success in a single trial
The probability of failure in a single trial (q = 1 – p)
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 shorthand
 Add or subtract multiple binomial computations
 Add 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:
Mean
μ = np
σ2 = npq Variance
σ = npq
Standard deviation
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}
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 L1
2) 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
P(x)
1
2
3
4
5
6
7
8
9
10
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
P(x) .00098
1
.0098
2
3
4
5
6
7
8
.044
.117
.205
.246
.205
.117
.044
9
.0098
10
.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