Section 4.2 - Binomial Distribution
Summer 2013 - Math 1040
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Roadmap
Binomial distributions describe repeated trials of an experiment that has a
success or failure result. We will see which symbols summarize the
distributions, compute the probability of a value for a random variable, and
how to summerize the random variable.
I
Binomial experiments.
I
Binomial probabilities.
I
Mean, variance, and standard deviations.
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Binomial Experiments
Binomial experiments must satisfy:
1. Repeat a fixed number of independent trials.
2. Result in a success or a failure for each trial.
3. The probability for success is fixed.
4. Have a random variable x counting the number of successes.
In symbols,
1. n is the number of independent trials.
2. p is the probability of a success for a trial,
3. q = 1 − p is the probability of a failure for a trial.
4. x may have a value of 0, 1, 2, . . . , n.
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Examples
Binomial:
1. Select a card at random from a standard deck of cards. Note if the
card is an ace or not. You perform this experiment n = 10 times, with
replacement. The random variable is the total number of aces.
2. A transplant operation has a 38% chance of success. A doctor
performs the operation on four patients. The random variable
represents the number of successful transplant operations.
Not binomial:
1. Select a card at random from a standard deck of cards. Note if the
card is an ace or not. You perform this experiment n = 10 times,
without replacement. The random variable is the total number of
aces.
2. In a game of Battleship you pick a coordinate to fire a missile. You
have 100 targetable coordinates, and you win the game when you hit
and sink all parts of your opponent’s ships. The random variable is
the total number hits.
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Binomial Probabilties
The binomial probability formula uses the Multiplication Rule and
combinations.
The probability for exaclty x successes in n trials is
P(x) =n Cx p x q n−x =
n!
p x q n−x .
(n − x)!x!
Example Knee surgeory has a 75% chance of success. The surgeory is
performed on three patients. The probability for exactly two successes is:
P(2) =
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3 2 1 1
3!
4
(3−2)!2! 4
1
9
= 3 16
4
= 27
≈
0.422
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Example
All probabilities for the knee surgeory are P(0), P(1), P(2), P(3) :
P(0) = 3 C0 (0.75)0 (0.25)3 = 1(0.75)0 (0.25)3 ≈ 0.0156
P(1) = 3 C1 (0.75)1 (0.25)2 = 3(0.75)1 (0.25)2 ≈ 0.1406
P(2) = 3 C2 (0.75)2 (0.25)1 = 3(0.75)2 (0.25)1 ≈ 0.4219
P(3) = 3 C3 (0.75)3 (0.25)0 = 1(0.75)3 (0.25)0 ≈ 0.4219
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Pascal’s Triangle
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Histograms
Steps for graphing a binomial distribution:
1. Label a horizontal numberline 0 to n.
2. Make the width of each bar one unit, drawing the center of the bar
over the number.
3. Make the area of each bar equal to its probability.
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Histogram
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Histogram
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Histogram
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Mean, variance, and standard deviation
Simpler formulas for a binomial distribution mean and variance /standard
deviation are:
µ = np
and
σ 2 = npq and σ =
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npq
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Example
Example In Seattle, WA, about 15.6% of the days in July are rainy. Find
the mean, variance, and standard deviation for the number of rainy days in
one random week.
n = 7 and p = 0.156 and q = 0.844.
µ = (7)(0.156) = 1.092
σ 2 = (7)(0.156)(0.844) ≈ 0.9216
σ=
p
(7)(0.156)(0.844) ≈ 0.96
On average, there are 1.092 rainy days in a week in July. It would be
unusual if more than 2 standard deviation days above the mean rain:
1.092 + (2)(0.96) = 3.012 days rain per week.
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Assignment
Assignment:
1. Read pages 202 - 210.
2. Exercises 1 - 33 odd, page 211.
3. Read the set up for multinomial experiments on page 215.
Vocabulary: binomial experiment, binomial probability formula and
distribution
Understand: Determine if a probability experiment is binomial, how to
find probabilities for a probability distribution table, their graphs, and the
mean, variance, and standard deviation of a binomial probability
distribution.
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