Additive Rules

• Theorem 2.10:
  

• Corollary 1:
  

Figure 1: Additive rule of probability.

• Corollary 2:
  
• Corollary 3:
  

• Theorem 2.11: (an extension of Theorem 2.10)
  
• Theorem 2.12:
  Proof : Since $A\cup A'= S$ and $A \cap A' = \varnothing$, then
  
  $$1 = P(S) = P(A \cup A') = P(A) + P(A')$$

• Example 2.32: The probability that the production procedure meets specification ( $2000 \pm 10~mm$) is known to be 0.99. Small cable is just as likely to be defective as large cable.
  • What is the probability that a cable selected randomly is too large?
    Let M be the event that a cable meets spec. Let $S$ and $L$ be the events that the cable is too small and too large, respectively. Then
    $$P(M) = 0.99~ and~ P(S) = P(L) = (1-0.99)/2 = 0.0005$$

  • What is the probability that a cable selected randomly is larger than 1990 mm?
    $$P(X \geq 1990) = 1-P(S) = 0.995$$
    where $X$ is the length of a randomly selected cable.