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 23:52, 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 r1 = rred = 0, r2 = rgreen = 2, and r3 = rblue = 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, p1 = pred = 2 / 9, p2 = pgreen = 1 / 3, and p3 = pblue = 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,cdots,k. Let pi be the probability of a trial resulting in the ith category, where . Let Ni be the number of trials resulting in the ith 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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