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A fundamental problem is to determine the probability of a logical (Boolean) combination of a finite class of events, when the probabilities of certain other combinations are known. If we partition an event F into component events whose probabilities can be determined, then the additivity property implies the probability of F is the sum of these component probabilities. Frequently, the event F is a Boolean combination of members of a finite class -- say {A, B, C} or {A, B, C,D}. For each such finite class, there is a fundamental partition determined by the class. The members of this partition are called minterms. Any Boolean combination of members of the class can be expressed as the disjoint union of a unique subclass of the minterms. If the probability of every minterm in this subclass can be determined, then by additivity the probability of the Boolean combination is determined. An important geometric aid to analysis is the minterm map, which has spaces for minterms in an orderly arrangement.

Introduction

A fundamental problem in elementary probability is to find the probability of a logical (Boolean) combination of a finite class of events, when the probabilities ofcertain other combinations are known. If we partition an event F into component events whose probabilities can be determined, then the additivity property implies the probability of F is the sum of these component probabilities. Frequently, the event F is a Boolean combination of members of a finite class– say, $\left\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C\right\}$ or $\left\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C,\phantom{\rule{0.166667em}{0ex}}D\right\}$ . For each such finite class, there is a fundamental partition determined by the class. The members of this partition are called minterms . Any Boolean combination of members of the class can be expressed as the disjoint union of a unique subclass of the minterms. If the probability of every mintermin this subclass can be determined, then by additivity the probability of the Boolean combination is determined. We examine these ideas in more detail.

Partitions and minterms

To see how the fundamental partition arises naturally, consider first the partition of the basic space produced by a single event A .

$\Omega =A\bigvee {A}^{c}$

Now if B is a second event, then

$A=AB\bigvee A{B}^{c}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{A}^{c}={A}^{c}B\bigvee {A}^{c}{B}^{c},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{so}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\Omega ={A}^{c}{B}^{c}\bigvee {A}^{c}B\bigvee A{B}^{c}\bigvee AB$

The pair $\left\{A,\phantom{\rule{0.166667em}{0ex}}B\right\}$ has partitioned Ω into $\left\{{A}^{c}{B}^{c},\phantom{\rule{0.166667em}{0ex}}{A}^{c}B,\phantom{\rule{0.166667em}{0ex}}A{B}^{c},\phantom{\rule{0.166667em}{0ex}}AB\right\}$ . Continuation is this way leads systematically to a partition by three events $\left\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C\right\}$ , four events $\left\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C,\phantom{\rule{0.166667em}{0ex}}D\right\}$ , etc.

We illustrate the fundamental patterns in the case of four events $\left\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C,\phantom{\rule{0.166667em}{0ex}}D\right\}$ . We form the minterms as intersections of members of the class, with various patterns of complementation.For a class of four events, there are ${2}^{4}=16$ such patterns, hence 16 minterms. These are, in a systematic arrangement,

 ${A}^{c}{B}^{c}{C}^{c}{D}^{c}$ ${A}^{c}B{C}^{c}{D}^{c}$ $A{B}^{c}{C}^{c}{D}^{c}$ $AB{C}^{c}{D}^{c}$ ${A}^{c}{B}^{c}{C}^{c}D$ ${A}^{c}B{C}^{c}D$ $A{B}^{c}{C}^{c}D$ $AB{C}^{c}D$ ${A}^{c}{B}^{c}C\phantom{\rule{0.277778em}{0ex}}{D}^{c}$ ${A}^{c}BC\phantom{\rule{0.277778em}{0ex}}{D}^{c}$ $A{B}^{c}C\phantom{\rule{0.277778em}{0ex}}{D}^{c}$ $ABC\phantom{\rule{0.277778em}{0ex}}{D}^{c}$ ${A}^{c}{B}^{c}C\phantom{\rule{0.277778em}{0ex}}D$ ${A}^{c}BC\phantom{\rule{0.277778em}{0ex}}D$ $A{B}^{c}C\phantom{\rule{0.277778em}{0ex}}D$ $ABC\phantom{\rule{0.277778em}{0ex}}D$

No element can be in more than one minterm, because each differs from the others by complementation of at least one member event. Eachelement ω is assigned to exactly one of the minterms by determining the answers to four questions:

Is it in A ? Is it in B ? Is it in C ? Is it in D ?

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