# AP Statistics Curriculum 2007 Distrib Multinomial

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The multinomial experiments (and multinomial distribtuions) directly extend the their [[AP_Statistics_Curriculum_2007_Distrib_Binomial |bi-nomial counterparts]]. | The multinomial experiments (and multinomial distribtuions) directly extend the their [[AP_Statistics_Curriculum_2007_Distrib_Binomial |bi-nomial counterparts]]. | ||

- | + | ===Multinomial experiments=== | |

- | ** | + | A multinomial experiment is an experiment that has the following properties: |

+ | * The experiment consists of '''k repeated trials'''. | ||

+ | * Each trial has a '''discrete''' number of possible outcomes. | ||

+ | * On any given trial, the probability that a particular outcome will occur is '''constant'''. | ||

+ | * The trials are '''independent'''; that is, the outcome on one trial does not affect the outcome on other trials. | ||

- | + | ====Examples of Multinomial experiments==== | |

- | + | * Suppose we have an urn containing 9 marbles. Two are red, three are green, and four are blue (2+3+4=9). We randomly select 5 marbles from the urn, ''with replacement''. What is the probability (''P(A)'') of the event ''A={selecting 2 green marbles and 3 blue marbles}''? | |

- | + | ||

- | + | ||

- | + | ||

- | + | *To solve this problem, we apply the multinomial formula. We know the following: | |

- | + | ** The experiment consists of 5 trials, so k = 5. | |

- | + | ** The 5 trials produce 0 red, 2 green marbles, and 3 blue marbles; so <math>r_1=r_{red} = 0</math>, <math>r_2=r_{green} = 2</math>, and <math>r_3=r_{blue} = 3</math>. | |

+ | ** For any particular trial, the probability of drawing a red, green, or blue marble is 2/9, 3/9, and 5/9, respectively. Hence, <math>p_1=p_{red} = 2/9</math>, <math>p_2=p_{green} = 1/3</math>, and <math>p_3=p_{blue} = 5/9</math>. | ||

+ | |||

+ | Plugging these values into the multinomial formula we get the probability of the event of interest to be: | ||

+ | |||

+ | : <math>P(A) = {5\choose 0, 2, 3}p_1^{r_1}p_2^{r_2}p_3^{r_3}</math> | ||

+ | |||

+ | : <math>P(A) = {5! \over 0!\times 2! \times 3! }\times (2/9)^0 \times (1/3)^2\times (5/9)^3=0.19052.</math> | ||

+ | |||

+ | Thus, if we draw 5 marbles with replacement from the urn, the probability of drawing no red , 2 green, and 3 blue marbles is ''0.19052''. | ||

===Synergies between Binomial and Multinomial processes/probabilities/coefficients=== | ===Synergies between Binomial and Multinomial processes/probabilities/coefficients=== |

## Revision as of 23:50, 4 March 2008

## Contents |

## General Advance-Placement (AP) Statistics Curriculum - Multinomial Random Variables and Experiments

The multinomial experiments (and multinomial distribtuions) directly extend the their bi-nomial counterparts.

### Multinomial experiments

A multinomial experiment is an experiment that has the following properties:

- The experiment consists of
**k repeated trials**. - Each trial has a
**discrete**number of possible outcomes. - On any given trial, the probability that a particular outcome will occur is
**constant**. - The trials are
**independent**; that is, the outcome on one trial does not affect the outcome on other trials.

#### Examples of Multinomial experiments

- Suppose we have an urn containing 9 marbles. Two are red, three are green, and four are blue (2+3+4=9). We randomly select 5 marbles from the urn,
*with replacement*. What is the probability (*P(A)*) of the event*A={selecting 2 green marbles and 3 blue marbles}*?

- To solve this problem, we apply the multinomial formula. We know the following:
- The experiment consists of 5 trials, so k = 5.
- The 5 trials produce 0 red, 2 green marbles, and 3 blue marbles; so
*r*_{1}=*r*_{red}= 0,*r*_{2}=*r*_{green}= 2, and*r*_{3}=*r*_{blue}= 3. - For any particular trial, the probability of drawing a red, green, or blue marble is 2/9, 3/9, and 5/9, respectively. Hence,
*p*_{1}=*p*_{red}= 2 / 9,*p*_{2}=*p*_{green}= 1 / 3, and*p*_{3}=*p*_{blue}= 5 / 9.

Plugging these values into the multinomial formula we get the probability of the event of interest to be:

Thus, if we draw 5 marbles with replacement from the urn, the probability of drawing no red , 2 green, and 3 blue marbles is *0.19052*.

### Synergies between Binomial and Multinomial processes/probabilities/coefficients

- The Binomial vs. Multinomial
**Coefficients**

- The Binomial vs. Multinomial
**Formulas**

- The Binomial vs. Multinomial
**Probabilities**

### Example

Suppose we study N independent trials with results falling in one of k possible categories labeled 1,2,*c**d**o**t**s*,*k*. Let *p*_{i} be the probability of a trial resulting in the *i*^{th} category, where . Let *N*_{i} be the number of trials resulting in the *i*^{th} category, where .

For instance, suppose we have 9 people arriving at a meeting according to the following information:

- P(by Air) = 0.4, P(by Bus) = 0.2, P(by Automobile) = 0.3, P(by Train) = 0.1

- Compute the following probabilities

- P(3 by Air, 3 by Bus, 1 by Auto, 2 by Train) = ?
- P(2 by air) = ?

### SOCR Multinomial Examples

### References

- SOCR Home page: http://www.socr.ucla.edu

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