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Binomial Distributions
Calculating the Probability of Success
Contents
1. How to identify binomial distributions.
2. How to calculate binomial probabilities.
3. When to use Normal approximations for binomial distributions.
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1. How to identify binomial distributions
Identification
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Binomial Distribution
Discrete random variable
Define X
S={0, 1, 2, …}
Binomial setting
XB(n, p)
Key idea: Count success!
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The Binomial Setting
1. “Success” or “Failure.”
2. Probability of success same for each trial.
3. Trials independent.
4. Fixed number of trials.
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Characteristics
XB(n, p)
Expected Value:
Variance:
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( )E X npX
2( ) (1 )V X np pX
2. How to Calculate Binomial Probabilities
Calculations
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Probability Calculations
Where:
k is the desired count,
n is the fixed number of trials,
p is the probability of success, and
(1-p) is the probability of failure.
( ) (1 )n k n kP X k p pk
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Example
What is the probability of tossing a fair coin five times and getting exactly three heads?
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Check for Binomial Setting
1. Success is flipping a head;failure is flipping a tail.
2. The probability of flipping heads on a fair coin is 50% each time.
3. Each flip is independent.
4. There is a fixed number of trials.
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Define Values
In our example:
k = 3
n = 5
p = 0.5 &
(1-p) = 0.5
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Calculations
5! 3 2( 3) (0.5) (0.5)3!2!
P X
5 3 2( 3) (0.5) (0.5)3
P X
54321 3 2( 3) (0.5) (0.5)32121
P X
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More Calculations
52( 3) (0.125)(0.25)1
P X
( 3) (10)(0.125)(0.25)P X
( 3) 0.3125P X
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Interpretation
There is about a 31% chance of flipping a fair coin 5 times and getting exactly 3 heads.
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Binomial Distribution
Using similar calculations,we can find each probability:
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X 0 1 2 3 4 5
P(X) 0.031 0.156 0.313 0.313 0.156 0.031
3. When to use Normal approximations.
Normal Approximations
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Normal Approximations
If n is large enough,
XB(n, p) XN(,).
Follow two “rules of thumb:”
1.np 10, &
2.N(1-p) 10
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The End
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