AP Statistics Curriculum 2007 Prob Rules

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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

\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

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