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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