AP Statistics Curriculum 2007 Prob Rules

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For events ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub> in a probability space (S,P), the probability of the union for ''n=2'' is
For events ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub> in a probability space (S,P), the probability of the union for ''n=2'' is
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:<math>\mathbb{P}(A_1\cup A_2)=\mathbb{P}(A_1)+\mathbb{P}(A_2)-\mathbb{P}(A_1\cap A_2),</math>
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:<math>P(A_1\cup A_2)=P(A_1)+P(A_2)-P(A_1\cap A_2),</math>
For ''n=3'',
For ''n=3'',
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:<math>\begin{align}\mathbb{P}(A_1\cup A_2\cup A_3)&=\mathbb{P}(A_1)+\mathbb{P}(A_2)+\mathbb{P}(A_3)\\
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:<math>P(A_1\cup A_2\cup A_3)=P(A_1)+P(A_2)+P(A_3) -P(A_1\cap A_2)-P(A_1\cap A_3)-P(A_2\cap A_3)+P(A_1\cap A_2\cap A_3)</math>
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&\qquad-\mathbb{P}(A_1\cap A_2)-\mathbb{P}(A_1\cap A_3)-\mathbb{P}(A_2\cap A_3)+\mathbb{P}(A_1\cap A_2\cap A_3)
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\end{align}</math>
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In general, for any ''n'',
In general, for any ''n'',
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:<math>\begin{align}
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:<math>P(\bigcup_{i=1}^n A_i) =\sum_{i=1}^n P(A_i)
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\mathbb{P}\biggl(\bigcup_{i=1}^n A_i\biggr) & {} =\sum_{i=1}^n \mathbb{P}(A_i)
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-\sum_{i,j\,:\,i<j}P(A_i\cap A_j) +\sum_{i,j,k\,:\,i<j<k}P(A_i\cap A_j\cap A_k)- \cdots\cdots\ \pm P(\bigcap_{i=1}^n A_i).</math>
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-\sum_{i,j\,:\,i<j}\mathbb{P}(A_i\cap A_j) \\
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&\qquad+\sum_{i,j,k\,:\,i<j<k}\mathbb{P}(A_i\cap A_j\cap A_k)-\ \cdots\cdots\ \pm \mathbb{P}\biggl(\bigcap_{i=1}^n A_i\biggr),
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\end{align}</math>
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This can also be written in closed form as
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:<math>\mathbb{P}\biggl(\bigcup_{i=1}^n A_i\biggr)  =\sum_{k=1}^n (-1)^{k-1}\sum_{\scriptstyle I\subset\{1,\ldots,n\}\atop\scriptstyle|I|=k} \mathbb{P}\biggl(\bigcap_{i\in I} A_i\biggr),</math>
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-
 
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where the last sum runs over all subsets ''I'' of the indices 1, ..., ''n'' which contain exactly ''k'' elements.
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=== Multiplication Rule===
=== Multiplication Rule===

Revision as of 03:54, 29 January 2008

Contents

General Advance-Placement (AP) Statistics Curriculum - Probability Theory Rules

Addition Rule

The probability of a union, also called the Inclusion-Exclusion principle allows us to compute probabilities of composite events represented as unions (i.e., sums) of simpler events.

Venn Diagrams

For events A1, ..., An in a probability space (S,P), the probability of the union for n=2 is

P(A_1\cup A_2)=P(A_1)+P(A_2)-P(A_1\cap A_2),

For n=3,

P(A_1\cup A_2\cup A_3)=P(A_1)+P(A_2)+P(A_3) -P(A_1\cap A_2)-P(A_1\cap A_3)-P(A_2\cap A_3)+P(A_1\cap A_2\cap A_3)

In general, for any n,

P(\bigcup_{i=1}^n A_i) =\sum_{i=1}^n P(A_i) -\sum_{i,j\,:\,i<j}P(A_i\cap A_j) +\sum_{i,j,k\,:\,i<j<k}P(A_i\cap A_j\cap A_k)- \cdots\cdots\ \pm P(\bigcap_{i=1}^n A_i).

Multiplication Rule

Model Validation

Checking/affirming underlying assumptions.

  • TBD

Computational Resources: Internet-based SOCR Tools

  • TBD

Examples

Computer simulations and real observed data.

  • TBD

Hands-on activities

Step-by-step practice problems.

  • TBD

References

  • TBD

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