2. Probability （概率）
2.1 Sample Space 样本空间
statistical experiment (random experiment)
----repeating
----more than one outcome
----know all the outcomes, but don’t predict which outcome will be occur
example:
toss an honest coin---- In this experiment there are only two possible outcomes:
{head}, {tail}
toss two honest coins---- In this experiment there are 4 possible outcomes:
{H, H}, {H, T}, {T, H}, {T, T}
toss three honest coins---- what is the possible outcomes?
Definition 2.1.1 The set of all possible outcomes of a statistical experiment is called the sample space. |
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Each outcome in a sample space is called a sample point of the sample space. Example 2.1.1 Consider the experiment of tossing a die. If we are interested in the number that shows on the top face, the sample space would be
If we are interested only in whether the numbers is even or odd, the sample space is simply
Example 2.1.3 An experiment consists of flipping a coin and then flipping it a second tim
e if a head occurs. If a tail occurs on the first flip then a die is tossing once. To list the elements of the sample space providing the most information,
we construct a diagram of Fig 2.1.1, which is called a tree diagram. Now the various paths along the branches of the tree give the distinct sample points. Starting with the top left branch and moving to the right along the first path, we get the sample point HH, indicating the possibility that heads occurs on two successive flips of the coin. The possibility that coin will show a tail followed by a 4 on the toss of the die is indicated by T4. Thus the sample space is
Fig .2.1.1 Tree diagram for Example 2.1.3
2.2 Events
Definition 2.2.1保护层垫块 An event is a subset of a sample space. |
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Example 2.2.1 Given the sample space . where is the life in hours of a certain bulb, we are interest in the event that a bulb burnt out before 200 hrs, i.e. the subset of .
Example 2.2.2 Assume that the unemployment rate of a region is between 0 and 15%，i.e. we have the sample space . If the event “unemployment rate is low” means that , then we have the subset of .
You may have known operation of subsets, i.e.
the complement of a subset（余集）,
the union of subset,(并集)
the difference of subset（差集）
intersection of subsets（交集）,
so we can say about the complement of an event, the union, difference and intersection of events.
certain event（必然事件）：
The sample space itself, is certainly an event, which is called a certain event, means that it always occurs in the experiment.
impossible event（不可能事件）：
The empty set, denoted by, is also an event, called an impossible event, means that it never occurs in the experiment.太阳能炉灶
Example 2.2.3 Consider the experiment of tossing a die, then
Let be the number that shows on the top face, then the event , is the certain event, i.e. .Then even is an irrational number, (irrational-无理数)is the impossible event, i.e..
Example 2.2.4 Consider the sample space consists of all positive integers less than 10, i.e.
Let be the event consisting of all even numbers and be the event consisting of numbers divisible by 3. Find , .
Solution We have 高硅铝合金Thus
The relationship between events and the corresponding sample space can be illustrated graphical by means of Venn diagrams. In a Venn diagram, we represent the sample space by a rectangle and represent events by circles drawn inside the rectangle.
Example 2.2.5 In Figure 2.2.1
= regions 1 and 2, = regions 1,2 ,3 ,4 ,5 and 7,
= regions 2, 6 and 7
Fig 2.2.1 Venn diagram of Example 2.2.5
The following list summarizes the rules of the operations of events.
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2.3 Probability of events
1.relative frequency --------probability
Considering an Example.
We plant 100 untreated cotton seeds.
If 49 seeds germinate, that is, if there are 49 success (by success in statistics we mean the occurrence of the event under discussion) in 100 trials, we say that the relative frequency of success is 0.49.
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