binomial distributions calculating the probability of success

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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