AP Statistics Curriculum 2007 Prob Rules

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:<math>P(A | B) ={P(A \cap B) \over P(B)}.</math>
:<math>P(A | B) ={P(A \cap B) \over P(B)}.</math>
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===Example===
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Here is the data on 400 Melanoma (skin cancer) Patients by Type and Site
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<center>
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{| class="wikitable" style="text-align:center; width:75%" border="1"
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|-
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| Site
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|-
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| Type || Head and Neck || Trunk || Extremities || Totals
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|-
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| Hutchinson's melanomic freckle || 22 || 2 || 10 || 34
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|-
 +
| Superficial || 16 || 54 || 115 || 185
 +
|-
 +
| Nodular || 19 || 33 || 73 || 125
 +
|-
 +
| Indeterminant || 11 || 17 || 28 || 56
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|-
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| Column Totals || 68 || 106 || 226 || 400
 +
|}
 +
</center>

Revision as of 04:08, 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

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\ +(-1)^{j+1} P(\bigcap_{i=1}^j A_i)+ \cdots\cdots\  +(-1)^{n+1} P(\bigcap_{i=1}^n A_i).


Conditional Probability

The conditional probability of A occurring given that B occurs is given by

P(A | B) ={P(A \cap B) \over P(B)}.

Example

Here is the data on 400 Melanoma (skin cancer) Patients by Type and Site

Site
Type Head and Neck Trunk Extremities Totals
Hutchinson's melanomic freckle 22 2 10 34
Superficial 16 54 115 185
Nodular 19 33 73 125
Indeterminant 11 17 28 56
Column Totals 68 106 226 400


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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