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5.20. Use the definition (5.15) of the probability of an event to prove the following basic facts about probability theory. (a) Let E and F be disjoint events. Then Pr(EU F) = Pr(E) + Pr(F). (b) Let E and F be events that need not be disjoint. Then Pr(EU F) = Pr(E) + Pr(F) − Pr(E^ F). (c) Let E be an event. Then Pr(Ec) = 1 − Pr(E). (d) Let E1, E2, E3 be events. Prove that Pr(E₁ ¯ E2 ¯ E3) = Pr(E₁) + Pr(E2) + Pr(E3) − Pr(E₁ ^ E₂) – Pr(E₁ ^ E3) — Pr(E2 ^ E3) + Pr(E₁ ^ E2 ^ E3). The formulas in (b) and (d) and their generalization to n events are known as the inclusion-exclusion principle.

5.20. Use the definition (5.15) of the probability of an event to prove the following basic facts about probability theory. (a) Let E and F be disjoint events. Then Pr(EU F) = Pr(E) + Pr(F). (b) Let E and F be events that need not be disjoint. Then Pr(EU F) = Pr(E) + Pr(F) − Pr(E^ F). (c) Let E be an event. Then Pr(Ec) = 1 − Pr(E). (d) Let E1, E2, E3 be events. Prove that Pr(E₁ ¯ E2 ¯ E3) = Pr(E₁) + Pr(E2) + Pr(E3) − Pr(E₁ ^ E₂) – Pr(E₁ ^ E3) — Pr(E2 ^ E3) + Pr(E₁ ^ E2 ^ E3). The formulas in (b) and (d) and their generalization to n events are known as the inclusion-exclusion principle.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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5.20. Use the definition (5.15) of the probability of an event to prove the following basic facts about probability theory. (a) Let E and F be disjoint events. Then Pr(EU F) = Pr(E) + Pr(F). (b) Let E and F be events that need not be disjoint. Then Pr(EUF) = Pr(E) + Pr(F) − Pr(E^F). (c) Let E be an event. Then Pr(Ec) = 1 − Pr(E). (d) Let E1, E2, E3 be events. Prove that Pr(E₁ U E2 U E3) = Pr(E₁) + Pr(E2) + Pr(E3) — Pr(E₁ ^ E₂) – Pr(E₁ E3) — Pr(E2 ~ E3) + Pr(E₁ E₂ E3). The formulas in (b) and (d) and their generalization to n events are known as the inclusion-exclusion principle.
Transcribed Image Text:5.20. Use the definition (5.15) of the probability of an event to prove the following basic facts about probability theory. (a) Let E and F be disjoint events. Then Pr(EU F) = Pr(E) + Pr(F). (b) Let E and F be events that need not be disjoint. Then Pr(EUF) = Pr(E) + Pr(F) − Pr(E^F). (c) Let E be an event. Then Pr(Ec) = 1 − Pr(E). (d) Let E1, E2, E3 be events. Prove that Pr(E₁ U E2 U E3) = Pr(E₁) + Pr(E2) + Pr(E3) — Pr(E₁ ^ E₂) – Pr(E₁ E3) — Pr(E2 ~ E3) + Pr(E₁ E₂ E3). The formulas in (b) and (d) and their generalization to n events are known as the inclusion-exclusion principle.
Definition. A sample space (or set of outcomes) is a finite⁹ set . Each outcome we is assigned a probability Pr(w), where we require that the probability function Pr : Ω satisfy the following two properties: (a) 0≤ Pr(w) ≤ 1 for all w EN and (b) Σ Pr(ω) = 1. (5.14) WEN R Notice that (5.14) (a) corresponds to our intuition that every outcome has a probability between 0 (if it never occurs) and 1 (if it always occurs), while (5.14) (b) says that some outcome must occur, so contains all possible outcomes for the experiment. Definition. An event is any subset of 2. We assign a probability to an event ECN by setting (5.15) Pr(E) = Σ Pr(w). WEE In particular, Pr(Ø) = 0 by convention, and Pr(N) = 1 from (5.14)(b).
Transcribed Image Text:Definition. A sample space (or set of outcomes) is a finite⁹ set . Each outcome we is assigned a probability Pr(w), where we require that the probability function Pr : Ω satisfy the following two properties: (a) 0≤ Pr(w) ≤ 1 for all w EN and (b) Σ Pr(ω) = 1. (5.14) WEN R Notice that (5.14) (a) corresponds to our intuition that every outcome has a probability between 0 (if it never occurs) and 1 (if it always occurs), while (5.14) (b) says that some outcome must occur, so contains all possible outcomes for the experiment. Definition. An event is any subset of 2. We assign a probability to an event ECN by setting (5.15) Pr(E) = Σ Pr(w). WEE In particular, Pr(Ø) = 0 by convention, and Pr(N) = 1 from (5.14)(b).
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