experiment随机实验

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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.
• For example, 10 days out of 30 2 desktops were sold. 0 1 2 1 2 10
• From this we can construct 4 the probabilities of an event (i.e. the # of desktop sold on a given day)…
The list must be mutually exclusive（互斥） i.e. no two outcomes can occur at the same time. Die roll{ number less than 4 or even number} Die roll {odd number or even number}
• S = {O1, O2, …, Ok}
2007會計資訊系統計學(一)上課投影片
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Sample Space: S = {O1, O2,…,Ok}
O1 O2
Sample Space
a sample space of a random experiment is a list of all possible outcomes of the experiment. The outcomes must be mutually exclusive and exhaustive.
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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.
2007會計資訊系統計學(一)上課投影片
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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}
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
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.
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.
2007會計資訊系統計學(一)上課投影片
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
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.
• 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
2007會計資訊系統計學(一)上課投影片
• 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+
2007會計資訊系統計學(一)上課投影片
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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）.
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Relative Frequency Approach
• Bits & Bytes Computer Shop tracks the number of desktop computer systems it sells over a month (30 days): Desktops Sold # of Days
2007會計資訊系統計學(一)上課投影片
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Classical ALeabharlann Baiduproach
• If an experiment has n possible outcomes, this method would assign a probability of 1/n to each outcome.
• Experiment: Rolling a die Sample Space: S = {1, 2, 3, 4, 5, 6} Probabilities: Each sample point has a 1/6 chance of occurring.
• “There is a 40% chance Bits & Bytes will sell 3 desktops on any given day”
2007會計資訊系統計學(一)上課投影片
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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”
2007會計資訊系統計學(一)上課投影片
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Classical 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
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:
• 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).
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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.
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
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