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# Probability Calculus

(Redirected from Rule 3)

The Probability Calculus is a list of mathematical rules that describe how probabilities work. The probability of a sentence is a number between 0 and 1, and these rules describe how these numbers are assigned (for the meaning of these numbers, please see interpretations of probability).

There are many different ways to present these rules. We give two such presentations below. While the two lists look different, they are actually equivalent: following one set of rules forces you to follow the other, and vice versa. Unless explicitly stated, any reference to the rules will be to the Useful Presentation.

A good exercise is to see if you can show why following the rules of the Minimal Presentation forces you to to follow the rules of the Extended Presentation.

## The Minimal Presentation

The minimal presentation is, as the name suggests, the smallest amount of rules needed to completely describe how probabilities work. It is useful when we want to say things about the probability calculus, but not so useful for solving problems using the probability calculus. For that, we have another presentation below.

Minimal Rule 1: No probability is less than zero

Minimal Rule 2: If $A$ is a tautology, $P\phantom{\rule{0}{0ex}}r\left(A\right)=1$

Minimal Rule 3: If $A,B$ are mutually exclusive, then $P\phantom{\rule{0}{0ex}}r\left(A\vee B\right)=P\phantom{\rule{0}{0ex}}r\left(A\right)+P\phantom{\rule{0}{0ex}}r\left(B\right)$

Minimal Rule 4: For any sentences $A,B,P\phantom{\rule{0}{0ex}}r\left(A\wedge B\right)=P\phantom{\rule{0}{0ex}}r\left(A\right)\cdot P\phantom{\rule{0}{0ex}}r\left(B|A\right)$

(see conditional probability for a note about Rule 4).

## The Extended Presentation

The extended presentation is a larger list of rules than the minimal presentation which gives you more tools to use in solving problems. This presentation is best for solving problems with the probability calculus. The list begins with the four rules given in the minimal presentation. Using those rules we can prove the rest of the rules presented here.

Rule 1: No probability is less than zero

Rule 2: If $A$ is a tautology, $P\phantom{\rule{0}{0ex}}r\left(A\right)=1$

Rule 3: If $A,B$ are mutually exclusive, then $P\phantom{\rule{0}{0ex}}r\left(A\vee B\right)=P\phantom{\rule{0}{0ex}}r\left(A\right)+P\phantom{\rule{0}{0ex}}r\left(B\right)$

Rule 4: For any sentences $A,B,P\phantom{\rule{0}{0ex}}r\left(A\wedge B\right)=P\phantom{\rule{0}{0ex}}r\left(A\right)\cdot P\phantom{\rule{0}{0ex}}r\left(B|A\right)$

Rule 5 (Negation Rule): For any sentence $A,P\phantom{\rule{0}{0ex}}r\left(¬A\right)=1-P\phantom{\rule{0}{0ex}}r\left(A\right)$

• Proof: $A\vee ¬A$ is a tautology. So $P\phantom{\rule{0}{0ex}}r\left(A\vee ¬A\right)=P\phantom{\rule{0}{0ex}}r\left(A\right)+P\phantom{\rule{0}{0ex}}r\left(¬A\right)=1$ Subtracting $P\phantom{\rule{0}{0ex}}r\left(A\right)$ from both sides we get $P\phantom{\rule{0}{0ex}}r\left(¬A\right)=1-P\phantom{\rule{0}{0ex}}r\left(A\right)$.

Rule 6 (Contradiction Rule): If $A$ is a contradiction, $P\phantom{\rule{0}{0ex}}r\left(A\right)=0$

• Proof: Suppose $A$ is a contradiction. Then $¬A$ is a tautology. So $P\phantom{\rule{0}{0ex}}r\left(¬A\right)=1$ (by Rule 2). By Rule 5 $P\phantom{\rule{0}{0ex}}r\left(¬A\right)=1-P\phantom{\rule{0}{0ex}}r\left(A\right)$. So $P\phantom{\rule{0}{0ex}}r\left(A\right)=0$.

Rule 7 (Logical Equivalence Rule): If $A,B$ are logically equivalent, then $P\phantom{\rule{0}{0ex}}r\left(A\right)=P\phantom{\rule{0}{0ex}}r\left(B\right)$

• Proof: Suppose $A,B$ are logically equivalent. So $A\vee ¬B$ is a tautology. So $P\phantom{\rule{0}{0ex}}r\left(A\vee ¬B\right)=1$. Since $A,B$ are logically equivalent it also follows that $A,¬B$ are mutually exclusive. So $1=P\phantom{\rule{0}{0ex}}r\left(A\vee ¬B\right)=P\phantom{\rule{0}{0ex}}r\left(A\right)+P\phantom{\rule{0}{0ex}}r\left(¬B\right)=P\phantom{\rule{0}{0ex}}r\left(A\right)+1-P\phantom{\rule{0}{0ex}}r\left(B\right)$. Subtracting $1$ from both sides and adding $P\phantom{\rule{0}{0ex}}r\left(B\right)$ to both sides we get $P\phantom{\rule{0}{0ex}}r\left(A\right)=P\phantom{\rule{0}{0ex}}r\left(B\right)$.

Rule 8 (Generalized Disjunction Rule): For any sentences $A,B,P\phantom{\rule{0}{0ex}}r\left(A\vee B\right)=P\phantom{\rule{0}{0ex}}r\left(A\right)+P\phantom{\rule{0}{0ex}}r\left(B\right)-P\phantom{\rule{0}{0ex}}r\left(A\wedge B\right)$

• Proof: $A\vee B$ is logically equivalent to $A\vee \left(¬A\wedge B\right)$. So $P\phantom{\rule{0}{0ex}}r\left(A\vee B\right)=P\phantom{\rule{0}{0ex}}r\left(A\vee \left(¬A\wedge B\right)\right)=P\phantom{\rule{0}{0ex}}r\left(A\right)+P\phantom{\rule{0}{0ex}}r\left(¬A\wedge B\right)$ by rule 3. Now $B$ is logically equivalent to $\left(A\wedge B\right)\vee \left(¬A\wedge B\right)$. So $P\phantom{\rule{0}{0ex}}r\left(B\right)=P\phantom{\rule{0}{0ex}}r\left(\left(A\wedge B\right)\vee \left(¬A\wedge B\right)\right)$. By rule 3 $P\phantom{\rule{0}{0ex}}r\left(B\right)=P\phantom{\rule{0}{0ex}}r\left(A\vee B\right)+P\phantom{\rule{0}{0ex}}r\left(¬A\vee B\right)$ So $P\phantom{\rule{0}{0ex}}r\left(¬A\wedge B\right)=P\phantom{\rule{0}{0ex}}r\left(B\right)-P\phantom{\rule{0}{0ex}}r\left(A\wedge B\right)$. Substituting this equation into $P\phantom{\rule{0}{0ex}}r\left(A\vee B\right)=P\phantom{\rule{0}{0ex}}r\left(A\right)+P\phantom{\rule{0}{0ex}}r\left(¬A\wedge B\right)$ we get $P\phantom{\rule{0}{0ex}}r\left(A\vee B\right)=P\phantom{\rule{0}{0ex}}r\left(A\right)+P\phantom{\rule{0}{0ex}}r\left(B\right)-P\phantom{\rule{0}{0ex}}r\left(A\wedge B\right)$.

Rule 9 (Special Conjunction Rule): If $A,B$ are independent, then $P\phantom{\rule{0}{0ex}}r\left(A\wedge B\right)=P\phantom{\rule{0}{0ex}}r\left(A\right)\cdot P\phantom{\rule{0}{0ex}}r\left(B\right)$

• Proof: We know that $P\phantom{\rule{0}{0ex}}r\left(A\wedge B\right)=P\phantom{\rule{0}{0ex}}r\left(A\right)\cdot P\phantom{\rule{0}{0ex}}r\left(B|A\right)$. If $A,B$ are independent then $P\phantom{\rule{0}{0ex}}r\left(B|A\right)=p\phantom{\rule{0}{0ex}}r\left(B\right)$. Substitute accordingly.

## Consequences of These Rules

Using these rules, we can solve simple probability problems. We can also develop other interesting and useful tools for dealing with probabilities, such as the total probability theorem, the rule for a disjunction of independent sentences, and Bayes' theorem.