Statistics Class 10 2/29/2012

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Statistics Class 10

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Quiz 8

When playing roulette at the Bellagio casino in Las Vegas, a gambler is trying to decide whether to bet $5 on the number 13 or to bet $5 that the outcome is any one of these five possibilities: 0 or 00 or 1 or 2 or 3. From Example 8, we know that the expected value of the $5 bet for a single

number is -26₵. For the $5 bet that the outcome is 0 or 00 or 1 or 2 or 3, there is a probability of 5/38 of making a net profit of $30 and a 33/38 probability of losing $5.

a. Find the expected value for the $5 bett that the outcome is 0 or 00 or 1 or 2 or 3.

b. Which bet is better: A $5 bet on the number 13 or a $5 bet that the outcome is 0 or 00 or 1 or 2 or 3? Why?

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Binomial Probability Distributions

A binomial probability distribution results from a procedure that meets all the following requirements.

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Binomial Probability Distributions

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Binomial Probability Distributions

1. The procedure has a fixed number of trials.

2. The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.)

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Binomial Probability Distributions

3. Each trial must have all outcomes classified into two categories

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Binomial Probability Distributions

3. Each trial must have all outcomes classified into two categories

(commonly referred to as success and failure).

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Binomial Probability Distributions

Note on Independence

Often when selecting a sample we do so without replacement. This means that our events are dependent, and violate rule 2 of the binomial

probability distribution. However we can use the 5% guideline for

cumbersome calculations, and treat dependent events independent as long as the sample size is no more than 5% of the population size.

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Binomial Probability Distributions

Notation for a Binomial Probability Distribution

S and F (success and failure) denote the two possible categories of outcomes.

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Binomial Probability Distributions

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Binomial Probability Distributions

• P(S)=p p=probability of success) • P(F)=q q=probability of failure)

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Binomial Probability Distributions

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Binomial Probability Distributions

• n denotes the fixed number of trials

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Binomial Probability Distributions

• x denotes a specific number of successes in n trials • p denotes the probability of success in one of the n

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Binomial Probability Distributions

trials

• q denotes the probability of failure in one of the n trials

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Binomial Probability Distributions

trials

• q denotes the probability of failure in one of the n trials

• P(x) denotes the probability of getting exactly x successes among the n trials

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Binomial Probability Distributions

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Binomial Probability Distributions

Binomial Probability Formula

In a binomial Probability distribution, probabilities can be calculated by using the binomial probability formula.

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Binomial Probability Distributions

First recall/learn: Factorial symbol (!) denotes the product of decreasing powers of positive whole numbers.

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Binomial Probability Distributions

First recall/learn: Factorial symbol (!) denotes the product of decreasing powers of positive whole numbers. So 4! = 4 ∙ 3 ∙ 2 ∙ 1 and 0! = 1.

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Binomial Probability Distributions

First recall/learn: Factorial symbol (!) denotes the product of decreasing powers of positive whole numbers. So 4! = 4 ∙ 3 ∙ 2 ∙ 1 and 0! = 1.

𝑃 𝑥 = 𝑛!

𝑛−𝑥 !𝑥! ∙ 𝑝𝑥 ∙ 𝑞𝑛−𝑥 for 𝑥 = 0,1,2, … , 𝑛

where

n=number of trials

x=number of success among n trials

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Binomial Probability Distributions

We are going to do example 3 three times! 1st_{by hand , 2}nd_{by excel , and 3}rd_{by table.}

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Binomial Probability Distributions

We are going to do example 3 three times! 1st_{by hand , 2}nd_{by excel , and 3}rd_{by table.}

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𝜇, 𝜎

2 , 𝑎𝑛𝑑 𝜎 for Binomial Distributions

Recall for a probability distribution that;

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𝜇, 𝜎

2 , 𝑎𝑛𝑑 𝜎 for Binomial Distributions

For a Binomial Distribution 𝜇, 𝜎2, 𝑎𝑛𝑑 𝜎 are given by the following formulas:

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𝜇, 𝜎

2 , 𝑎𝑛𝑑 𝜎 for Binomial Distributions

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𝜇, 𝜎

2 , 𝑎𝑛𝑑 𝜎 for Binomial Distributions

𝜇 = 𝑛 ∙ 𝑝 𝜎2 = 𝑛 ∙ 𝑝 ∙ 𝑞

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𝜇, 𝜎

2 , 𝑎𝑛𝑑 𝜎 for Binomial Distributions

𝜇 = 𝑛 ∙ 𝑝 𝜎2 = 𝑛 ∙ 𝑝 ∙ 𝑞 𝜎 = 𝑛 ∙ 𝑝 ∙ 𝑞

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𝜇, 𝜎

2 , 𝑎𝑛𝑑 𝜎 for Binomial Distributions

𝜇 = 𝑛 ∙ 𝑝 𝜎2 = 𝑛 ∙ 𝑝 ∙ 𝑞 𝜎 = 𝑛 ∙ 𝑝 ∙ 𝑞

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Statistics Class 10 2/29/2012