목차

• Basic Probability Concepts
  • Assessing Probability
  • EVENT
  • Probability
  • Exclusive Events
  • Computing
• Conditional Probabilities
• PLUS
• Bayes’ Theorem
• Counting Rules

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이 아이콘들을 누르시면 정답, 개념 부가 설명을 보실 수 있습니다:)

Basic Probability Concepts

• Probability(확률): the chance that an uncertain event will occur (always between 0 and 1)
• Impossible Event: an event that has no chance of occurring (probability = 0)
• Certain Event: an event that is sure to occur (probability = 1)

Assessing Probability

There are three approaches to assessing the probability of an uncertain event:

1) a priori

• based on prior knowledge of the process
• 즉, 아직 일어나지 않았지만 일어나기 전 짐작할 수 있는 확률
  

2) empirical probability

• 중요하고 잘 쓰임
• 경험적 확률이다
• 이미 벌어져있다. 즉 증거로 쓸수 있다는 것이다
  

3) subjective probability

EVENT

Each possible outcome of a variable is an event.

[Simple event]
• An event described by a single characteristic • e.g., A day in January from all days in 2015

[Joint event]
• An event described by two or more characteristics

• 즉, 동시에 일어나는 것
  • e.g. A day in January that is also a Wednesday from all days in 2015

[Complement of an event A]
• All events that are not part of event A

• 즉, 여집합이다.
• denoted A’
  • e.g., All days from 2015 that are not in January

[Sample Space]

• collection of all possible events
• 즉, 일어날 수 있는 경우의 수
• ex) 주사위는 6면이니 6이다

[Venn Diagram]

• 우리가 흔히 아는 벤다이어그램이다

[Contingency Tables]

• 오른쪽 맨 아래: Total Number Of Sample Space Outcome

[Decision Tree]

• 알고리즘에서와 그림이 많이 달라서 그냥 넘어가도 됨

Probability

[Simple Probability]

• probability of a simple event

[Joint Probability]

• occurrence of two or more events (joint event)
• 동시에 일어나는 사건에 대한 확률

Exclusive Events

[Mutually Exclusive Events]

• Mutually exclusive events: Events that cannot occur simultaneously
  • A와 B가 Mutually exclusive events라면 동시에 일어날 수 없다는 것을 의미한다

[Collectively Exhaustive Events]

• Collectively exhaustive events
  • One of the events must occur
  • The set of events covers the entire sample space
  • 주어진 경우를 제외하고는 없다!
  • 즉 주어진 보기가 전체를 줬다

Computing

[Joint Probabilities]

• 교집합
• 두 사건을 보두 만족하는 것을 구하면 된다
  

[Marginal Probabilities]

• Where B1, B2, …, Bk are k mutually exclusive and collectively exhaustive events

[Marginal & Joint Probabilities In A Contingency Table]

[General Addition Rule]

• General Addition Rule
  
• If A and B are mutually exclusive, then P(A and B) = 0, so the rule can be simplified:
  

Conditional Probabilities

probability of one event, given that another event has occurred
즉, B가 일어났을때 A가 일어날 확률이다.

Where P(A and B) = joint probability of A and B

• P(A) = marginal or simple probability of A
• P(B) = marginal or simple probability of B

PLUS

[Independence]
Two events are independent if and only if

• Events A and B are independent when the probability of one event is not affected by the fact that the other event has occurred
• 즉,A가 일어나는데 B가 아무 상관이 없다는 이야기다.

[Multiplication Rules]
Multiplication rule for two events A and B:

Note: If A and B are independent, then and the multiplication rule simplifies to
Independence에 의해서,

Bayes’ Theorem

• Bayes’ Theorem is used to revise previously calculated probabilities based on new information.
• 즉 이전에 계산된 확률을 수정하는데 사용
• 조건부 확률의 연장이다

where:

• Bi = ith event of k mutually exclusive and collectively exhaustive events
• A = new event that might impact P(Bi)

Counting Rules

즉 순열과 조합이다.

[Counting Rule 1]
상호 배타적(mutually exclusive)이고 집단적으로 철저한(collectively exhaustive events) 개의 사건 중 하나가 n개의 시험 각각에서 발생할 수 있는 경우,

가능한 결과의 수:
\(K^n\)

[Counting Rule 2]
1차 시행에서 \(k_1\) 이벤트가, 2차 시행에서 \(k_2\) 이벤트가, 그리고 n차 시행에서 \(k_n\) 이벤트가 있는 경우,

가능한 결과의 수:
\((k_1)(k_2)...(K^n)\)

[Counting Rule 3]
Permutations(순열): The number of ways of arranging X objects selected from n objects in order is( n개의 개체에서 선택한 X개의 개체를 순서대로 정렬하는 방법 수),

가능한 결과의 수:
\(nP_x = n!/(n-X)!\)

[Counting Rule 4]
Combinations(조합): The number of ways of selecting X objects from n objects, irrespective of order, is

가능한 결과의 수:
\(nC_x = n!/X!(n-X)!\)

문제

1. If two events are collectively exhaustive, what is the probability that one or the other occurs?

2. If two events are mutually exclusive and collectively exhaustive, what is the probability that both occur?

[TABLE 4]
In a meat packaging plant Machine A accounts for 60% of the plant’s output, while Machine B accounts for 40% of the plant’s output. In total, 4% of the packages are improperly sealed. Also, 3% of the packages are from Machine A and are improperly sealed.

3. Referring to Table 4, if a package selected at random is improperly sealed, the probability that it came from machine A is ________.

4. Referring to Table 4, if a package selected at random came from Machine B, the probability that it is properly sealed is ________.

5. Based on the report, during prime time, husbands are watching TV 60% of time. When the husband is watching TV, 40% of the time the wife is also watching. When the husband is not watching TV, 30% of the time the wife is watching TV. Find the probability that if the wife is watching TV, the husband is also watching TV.

6. A company has 2 machines that produce widgets. An older machine produces 23% defective widgets, while the new machine produces only 8% defective widgets. In addition, the new machine produces 3 times as many widgets as the older machine does. Given a randomly chosen widget was tested and found to be defective, what is the probability it was produced by the new machine?

7. There are only 4 empty rooms available in a student dormitory for eleven new freshmen. Each room is considered unique so that it matters who is being assigned to which room. How many different ways can those 4 empty rooms be filled one student per room?

8. A debate team of 4 is to be chosen from a class of 35. There are two twin brothers in the class.How many possible ways can the team be formed which will include only one of the twin brothers?

9. A debate team of 4 is to be chosen from a class of 35. There are two twin brothers in the class. How many possible ways can the team be formed which will not include any of the twin brothers?