experiment随机实验

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3
12
5
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Relative Frequency Approach
Desktops Sold # of Days Desktops Sold
0 1
1 2
1/30 = .03 2/30 = .07
2
3 4
10
12 5
10/30 = .33
12/30 = .40 5/30 = .17 ∑ = 1.00
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Probabilities
• List the outcomes of a random experiment…
•
This list must be exhaustive（周延） i.e. ALL possible outcomes included. Die roll {1,2,3,4,5} Die roll {1,2,3,4,5,6}
2007assical Approach
• Experiment: Rolling dice Sample Space: S = {2, 3, …, 12} What are the underlying, Probability Examples:
unstated assumptions??
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2007會計資訊系統計學(一)上課投影片
Approaches to Assigning Probabilities
• There are three ways to assign a probability, P(Oi), to an outcome, Oi, namely:
• Classical approach（古典法）: make certain assumptions (such as equally likely, independence) about situation. • Relative frequency（相對次數法）: assigning probabilities based on experimentation or historical data. • Subjective approach（主觀法）: Assigning probabilities based on the assignor’s judgment.
Chapter 6
Probability
2007會計資訊系統計學(一)上課投影片
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6.1 Assigning probabilities to Events
• Random experiment（隨機實驗）
– a random experiment is a process or course of action, whose outcome is uncertain.
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•
2007會計資訊系統計學(一)上課投影片
Sample Space
• A list of exhaustive and mutually exclusive outcomes is called a sample space（樣本空間） and is denoted by S. • The outcomes are denoted by O1, O2, …, Ok • Using notation from set theory, we can represent the sample space and its outcomes as:
• Examples
Experiment Flip a coin Exam Marks Outcomes Heads, Tails Numbers: 0, 1, 2, ..., 100
Assembly Time Course Grades
2007會計資訊系統計學(一)上課投影片
t > 0 seconds F, D, C, B, A, A+
• Roll of a die: S = {1, 2, 3, 4, 5, 6} Simple event: the number “3” will be rolled Complex event: an even number (one of 2, 4, or 6) will be rolled
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Events & Probabilities
• An event（事件） is a collection or set of one or more outcomes in a sample space. • An individual outcome of a sample space is called a simple event（簡單事件）, while • A collection of two or more outcomes is called a complex event（複合事件）.
P(Oi) represents the probability of outcome i
P( O ) 1
i 1 i
•
Probability of an event: The probability P(A) of event A is the sum of the probabilities assigned to the simple events contained in A.
Our objective is to determine P(A), the probability that event A will occur.
2007會計資訊系統計學(一)上課投影片
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Requirements of Probabilities
• • • Given a sample space S = {O1, O2, …, Ok}, the probabilities assigned to the outcome must satisfy these requirements: The probability of any outcome is between 0 and 1 i.e. 0 ≤ P(Oi) ≤ 1 for each i, and The sum of the probabilities of all the outcomes equals 1 i.e. P(O1) + P(O2) + … + P(Ok) = 1 k
1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7
P(2) = 1/36
1 2 3 4 5 6
6 7 8
P(6) = 5/36 P(10) = 3/36
2007會計資訊系統計學(一)上課投影片
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• E.g. weather forecasting’s “P.O.P.” • “Probability of Precipitation” (or P.O.P.) is defined in different ways by different forecasters, but basically it’s a subjective probability based on past observations combined with current weather conditions. • POP 60% – based on current conditions, there is a 60% chance of rain (say).
Simple Events Event（事件）
An event is any collection of one or more simple events The individual outcomes are called simple events. Simple events cannot be further decomposed into constituent outcomes.
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Events & Probabilities
• The probability of an event is the sum of the probabilities of the simple events that constitute the event.
• “There is a 40% chance Bits & Bytes will sell 3 desktops on any given day”
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Subjective Approach
• “In the subjective approach we define probability as the degree of belief that we hold in the occurrence of an event”
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6.1 Assigning probabilities to Events
• Performing the same random experiment repeatedly, may result in different outcomes, therefore, the best we can do is consider the probability of occurrence of a certain outcome. • To determine the probabilities we need to define and list the possible outcomes first.