3.2 Sample Spaces and Events

Chapter Three

Probability

Probability

3.1 The Concept of Probability

3.2 Sample Spaces and Events

3.3 Some Elementary Probability Rules

3.4 Conditional Probability and Independence

*3.5 Bayes’ Theorem

3.1 Probability Concepts

An experiment is any process of observation with an uncertain outcome.

The possible outcomes for an experiment are called the experimental outcomes.

Probability is a measure of the chance that an experimental outcome will occur when an experiment is carried out

Probability

If E is an experimental outcome, then P(E) denotes the probability that E will occur and

Conditions

If E can never occur, then P(E) = 0

If E is certain to occur, then P(E) = 1

The probabilities of all the experimental outcomes must sum to 1.

Interpretation: long-run relative frequency or subjective

3.2 The Sample Space

The sample space of an experiment is the set of all experimental outcomes.

Example 3.2: Genders of Two Children

Computing Probabilities of Events

An event is a set (or collection) of experimental outcomes.

The probability of an event is the sum of the probabilities of the experimental outcomes that belong to the event.

Example: Computing Probabilities

Example 3.4: Genders of Two Children

Note: Experimental Outcomes: BB, BG, GB, GG

All outcomes equally likely:

Probabilities: Equally Likely Outcomes

If the sample space outcomes (or experimental outcomes) are all equally likely, then the probability that an event will occur is equal to the ratio

Example: AccuRatings Case

Of 5528 residents sampled, 445 prefer KPWR

Estimated Share:P(KPWR) = 445/5528= 0.0805

Assuming 8,300,000 Los Angeles residents aged 12 or older:

Listeners = Population x Share = 8,300,000 x 0.08 = 668,100

3.3 Event Relations

The complement of an event A is the set of all sample space outcomes not in A.

Further

Union of A and B,

Elementary events that belong to either A or B (or both.)

Intersection of A and B,

Elementary events that belong to both A and B.

The Addition Rule for Unions

The probability that A or B (the union of A and B) will occur is

A and B are mutually exclusive if they have no sample space outcomes in common, or equivalently if

3.4 Conditional Probability

The probability of an event A, given that the event B has occurred is called the “conditional probability of A given B” and is denoted as . Further

Independence of Events

Two events A and B are said to be independent if and only if:

P(A|B) = P(A) or, equivalently,

P(B|A) = P(B)

The Multiplication Rule for Intersections

The probability that A and B (the intersection of A and B) will occur is

If A and B are independent, then the probability that A and B (the intersection of A and B) will occur is

Contingency Tables

Example: AccuRatings Case

Example 3.16: Estimating Radio Station Share by Daypart

5528 L.A. residents sampled.

2827 of residents sampled listen during some portion of the 6-10 a.m. daypart.

Of those, 201 prefer KIIS

KIIS Share for 6-10 a.m. daypart:

Bayes’ Theorem

S1, S2, …, Sk represents k mutually exclusive possible states of nature, one of which must be true.

P(S1), P(S2), …, P(Sk) represents the prior probabilities of the k possible states of nature.

If E is a particular outcome of an experiment designed to determine which is the true state of nature, then the posterior (or revised) probability of a state Si, given the experimental outcome E, is:

3.5 Bayes’ Theorem: An Example, AIDS Testing

Question: Suppose that a person selected randomly for testing, tests positive for AIDS. The test is known to be highly accurate (99.9% for people who have AIDS, 99% for people who don’t.) What is the probability that the person actually has AIDS?

Answer: Surprisingly, much lower than most of us would guess!

An Example, AIDS Testing (Continued)

Probability

3.1 The Concept of Probability

3.2 Sample Spaces and Events

3.3 Some Elementary Probability Rules

3.4 Conditional Probability and Independence

*3.5 Bayes’ Theorem

Chapter Three / 第三章 Probability / 機率

Probability / 機率

3.1 The Concept of Probability / 3.1 機率的概念

3.2 Sample Spaces and Events / 3.2 抽樣調查空間和大事

3.3 Some Elementary Probability Rules / 3.3 ，一些基本的機率規定

3.4 Conditional Probability and Independence / 3.4 條件概率和獨立

*3.5 Bayes’ Theorem / *3.5個 Bayes' 定理

3.1 Probability Concepts / 3.1 機率概念

An experiment is any process of observation with an uncertain outcome. / 一個實驗用一個不確定的結果是觀察的任何程序。

The possible outcomes for an experiment are called the experimental outcomes. / 給一個實驗的合理結果叫做實驗的結果。

Probability is a measure of the chance that an experimental outcome will occur when an experiment is carried out / 機率是對機會的衡量當一個實驗被執行的時候，一個實驗的結果將發生

Probability / 機率

If E is an experimental outcome, then P(E) denotes the probability that E will occur and / 如果 E 是一個實驗的結果, 那麼 P(E) 指示 E 將發生的機率和

Conditions / 情況

If E can never occur, then P(E) = 0 / 如果 E 能不曾發生, 然後 P(E)=0

If E is certain to occur, then P(E) = 1 / 如果 E 確定發生, 然後 P(E)=1

The probabilities of all the experimental outcomes must sum to 1. / 所有的實驗結果的機率一定要總計到 1. Interpretation: long-run relative frequency or subjective / 翻譯: 長期相對的次數或主觀的

3.2 The Sample Space / 3.2 樣品空間

The sample space of an experiment is the set of all experimental outcomes. / 實驗的樣品空間是所有實驗的結果的套。

Example 3.2: Genders of Two Children / 例子 3.2: 二個孩子的性別

Computing Probabilities of Events / 大事的電腦機率

An event is a set (or collection) of experimental outcomes. / 一個大事是一組實驗的結果。

The probability of an event is the sum of the probabilities of the experimental outcomes that belong to the event. / 大事的機率是屬於大事的實驗結果的機率的總數。

Example: Computing Probabilities / 例子: 電腦機率

Example 3.4: Genders of Two Children / 例子 3.4: 二個孩子的性別

Note: Experimental Outcomes: BB, BG, GB, GG / 注意:實驗的結果: BB ， BG ，億位元組, GG

All outcomes equally likely: / 所有的結果相等有可能的:

Probabilities: Equally Likely Outcomes / 機率: 相等有可能的結果

If the sample space outcomes (or experimental outcomes) are all equally likely, then the probability that an event will occur is equal to the ratio / 如果樣品空間結果 (或實驗的結果) 全部相等有可能, 然後機率一個大事將發生和比率相等

Example: AccuRatings Case / 例子: AccuRatings 個案

Of 5528 residents sampled, 445 prefer KPWR / 5528個居民抽樣調查,445 偏愛 KPWR

Estimated Share:P(KPWR) = 445/5528= 0.0805 / 估計佔有率:P(KPWR)=445/5528=0.0805

Assuming 8,300,000 Los Angeles residents aged 12 or older: / 傲慢的 8,300,000個洛杉磯居民年老的 12 或資深者: Listeners = Population x Share = 8,300,000 x 0.08 = 668,100 / Listeners= population x 分享 =8,300,000 x 0.08=668,100

3.3 Event Relations / 3.3 大事關係

The complement of an event A is the set of all sample space outcomes not in A. / 大事一的補足物是所有樣品空間結果的套不在一。

Further / 比較進一步的

Union of A and B, / 聯合一和 B,

Elementary events that belong to either A or B (or both.) / 屬於的基本的大事或一或 B(或兩者的.)

Intersection of A and B, / 十字路口一和 B,

Elementary events that belong to both A and B. / 屬於的基本的大事兩者的一和 B。

The Addition Rule for Unions / 為聯合的增添規則

The probability that A or B (the union of A and B) will occur is / 機率哪一一或 B(聯合一和 B) 將發生是

A and B are mutually exclusive if they have no sample space outcomes in common, or equivalently if / 一而且如果他們沒有樣品隔開共同的結果， B 是互斥的, 或相等地如果

3.4 Conditional Probability / 3.4 條件概率

The probability of an event A, given that the event B has occurred is called the “conditional probability of A given B” and is denoted as . Further / 被給大事 B 已經發生的大事一的機率叫做 " 給定 B 的條件概率 " 而且被指示當做 . 比較進一步的

Independence of Events / 大事的獨立

Two events A and B are said to be independent if and only if: / 二個大事一和 B 被說是獨立的如果而且只有當如果:

P(A|B) = P(A) or, equivalently, / P(一|B)= P(一) 或, 相等地, P(B|A) = P(B) / P(B|一)= P(B)

The Multiplication Rule for Intersections / 為十字路口的乘法規則

The probability that A and B (the intersection of A and B) will occur is / 機率哪一一和 B(十字路口一和 B) 將發生是

If A and B are independent, then the probability that A and B (the intersection of A and B) will occur is / 如果一和 B 是獨立的, 然後機率一和 B(十字路口一和 B) 將發生是

Contingency Tables / 或然關係表

Example: AccuRatings Case / 例子: AccuRatings 個案

Example 3.16: Estimating Radio Station Share by Daypart / 例子 3.16: 估計 Daypart 的廣播電台佔有率

5528 L.A. residents sampled. / 5528個洛杉磯居民抽樣調查。

2827 of residents sampled listen during some portion of the 6-10 a.m. daypart. / 被抽樣調查的居民中的 2827 在 6 早上 10 點 daypart 的一些部分期間聽。

Of those, 201 prefer KIIS / 那些,201 偏愛 KIIS

KIIS Share for 6-10 a.m. daypart: / 為 6 早上 10 點的 daypart 的 KIIS 佔有率:

Bayes’ Theorem / Bayes' 定理

S1, S2, …, Sk represents k mutually exclusive possible states of nature, one of which must be true. / S1 ， S2 ，…, Sk 代表 k 自然的互斥合理狀態,哪一個的其中之一一定要是真實的。

P(S1), P(S2), …, P(Sk) represents the prior probabilities of the k possible states of nature. / P(S1) ， P(S2) ，…, P(Sk) 代表 k 的原有機率自然的合理狀態。

If E is a particular outcome of an experiment designed to determine which is the true state of nature, then the posterior (or revised) probability of a state Si, given the experimental outcome E, is: / 如果 E 是設計決定自然的真實狀態是哪一個的實驗的一個特別結果, 然後被給實驗的結果 E 的國家 Si 的在後 (或者修訂的) 機率, 是:

3.5 Bayes’ Theorem: An Example, AIDS Testing / 3.5 Bayes' 定理: 一個例子, 愛滋病測驗

Question: Suppose that a person selected randomly for testing, tests positive for AIDS. The test is known to be highly accurate (99.9% for people who have AIDS, 99% for people who don’t.) What is the probability that the person actually has AIDS? / 問題: 推想一個人隨機地為測驗選擇，對於愛滋病的檢驗確定的. 檢驗被知道高度地正確的 (99.9% 為有的人愛滋病,99% 對於人不.) 機率是什麼人實際上有愛滋病?

Answer: Surprisingly, much lower than most of us would guess! / 答案:令人驚訝地, 比我們大部份下跌許多將會猜測!

An Example, AIDS Testing (Continued) / 一個例子, 愛滋病測驗 (繼續)

Probability / 機率 –

3.1 The Concept of Probability / 3.1 機率的概念

3.2 Sample Spaces and Events / 3.2 抽樣調查空間和大事

3.3 Some Elementary Probability Rules / 3.3 ，一些基本的機率規定

3.4 Conditional Probability and Independence / 3.4 條件概率和獨立

*3.5 Bayes’ Theorem / *3.5個 Bayes' 定理