AP Statistics Curriculum 2007 Prob Rules

From Socr

(Difference between revisions)
Jump to: navigation, search
Line 5: Line 5:
[[Image:500px-Inclusion-exclusion.svg.png|150px|thumbnail|right| [http://upload.wikimedia.org/wikipedia/commons/thumb/4/42/Inclusion-exclusion.svg/180px-Inclusion-exclusion.svg.png Venn Diagrams]]]
[[Image:500px-Inclusion-exclusion.svg.png|150px|thumbnail|right| [http://upload.wikimedia.org/wikipedia/commons/thumb/4/42/Inclusion-exclusion.svg/180px-Inclusion-exclusion.svg.png Venn Diagrams]]]
 +
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
 +
:<math>\mathbb{P}(A_1\cup A_2)=\mathbb{P}(A_1)+\mathbb{P}(A_2)-\mathbb{P}(A_1\cap A_2),</math>
 +
 +
For ''n=3'',
 +
 +
:<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)\\
 +
&\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)
 +
\end{align}</math>
 +
 +
In general, for any ''n'',
 +
 +
:<math>\begin{align}
 +
\mathbb{P}\biggl(\bigcup_{i=1}^n A_i\biggr) & {} =\sum_{i=1}^n \mathbb{P}(A_i)
 +
-\sum_{i,j\,:\,i<j}\mathbb{P}(A_i\cap A_j) \\
 +
&\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),
 +
\end{align}</math>
 +
 +
This can also be written in closed form as
 +
 +
:<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>
 +
 +
where the last sum runs over all subsets ''I'' of the indices 1, ..., ''n'' which contain exactly ''k'' elements.
=== Multiplication Rule===
=== Multiplication Rule===

Revision as of 03:50, 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.

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

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

For n=3,

\begin{align}\mathbb{P}(A_1\cup A_2\cup A_3)&=\mathbb{P}(A_1)+\mathbb{P}(A_2)+\mathbb{P}(A_3)\\
&\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)
\end{align}

In general, for any n,

\begin{align}
\mathbb{P}\biggl(\bigcup_{i=1}^n A_i\biggr) & {} =\sum_{i=1}^n \mathbb{P}(A_i)
-\sum_{i,j\,:\,i<j}\mathbb{P}(A_i\cap A_j) \\
&\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),
\end{align}

This can also be written in closed form as

\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),

where the last sum runs over all subsets I of the indices 1, ..., n which contain exactly k elements.

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



Translate this page:

(default)

Deutsch

Español

Français

Italiano

Português

日本語

България

الامارات العربية المتحدة

Suomi

इस भाषा में

Norge

한국어

中文

繁体中文

Русский

Nederlands

Ελληνικά

Hrvatska

Česká republika

Danmark

Polska

România

Sverige

Personal tools