Like article \ video can not be separated throwing dice or flip a coin both instances all talk of probability theory, because throwing dice indeed is the basis of probability theory generated gamblers who do not care about God in order to win money on the probability theory can only lead to breakthrough religious strangling, so we are here to throw the dice and toss a coin two examples.
Let's go over to the basic concepts:
- Randomized trial. In the process of throwing the dice, we note that, under the same conditions, every time we throw the dice, we still can not know when the last dice, which faces on the dice, but we continue to throw the dice by the process of discovery, the result is nothing more than one of 1,2,3,4,5,6 six points. Randomized trials:
1. Test may be repeatedly performed under the same conditions.
2. Test results of more than one, and the clear experiment in advance all possible outcomes,
You can not predict the results of which will appear before 3. Test.
- Sample space. Given a random event, consisting of a collection of all possible outcomes is called the sample space. In this example, throwing dice, the sample space s = {1,2,3,4,5,6}
- Sample points. Random events that may result called sample points. In this case, throwing dice, each point among the six points 1,2,3,4,5,6 are called sample points can be seen, the sample space by the sample points.
- Random events, a subset of the sample space to meet certain conditions, referred to as random events. For example, "appears even"
- The basic events a sample point
- Impossible event. Empty set, such as the emergence 0:00
- Random events occur. Random events that occur in a random set of events within
- Complete Event Group. The elements contain different from each other between events, just add up all event groups constitute the sample space Collection
There are several random event relationship (in fact, a collection of 1933, the Russian mathematician Andrei N. Kolmogorov established axiomatic system of probability theory, strictly defines the language of probability theory. As with other modern subjects like mathematics, probability theory axiomatic set theory based on the same system. axiomatic probability theory based on a few straightforward axiom, derived from the entire system of probability theory. this study axiomatic system can eliminate a lot of confusion in intuition. this axiom system the core is the "probability measure"):
- Inclusion relations: A⊂B That event B contains event A, event A occurs inevitably lead to the occurrence of an event B
- Equivalence relations: A = BA⊂B, B⊂A
- And relationships: A∪B event A and event B occur at least a
- Event product: A∩B i.e. events A and B A post event B occurs simultaneously. The so-called events occur simultaneously, due to random events is a subset of the sample space, that is, it consists of a series of sample points. Therefore, when the sample points intersect two events, the occurrence of the same sample points, i.e. the event A and event B occurs simultaneously.
- And set. D = A ∪ B
- Poor Event: A-BB and A does not occur occur.
- Inverse or opposite event event:
- Mutually exclusive events. Φ empty set is a collection that does not contain any elements. If the intersection of two sets is the empty set, i.e., M ∩ N = [Phi], then the two disjoint sets. In probability theory, the two events are not mutually exclusive intersect.
Operation events (actually a set of operations, and as an addition, a set of cross-union operations also have algorithms.):
- Commutative. A ∪ B = B ∪ A and A ∩ B = B ∩ A
- Associativity. ( A ∪ B ) ∪ C = A ∪ ( B ∪ C ) and ( A ∩ B ) ∩ C = A ∩ ( B ∩ C )
- Distributive law. ( A ∪ B ) ∩ C = ( A ∩ C ) ∪ ( B ∩ C ) and ( A ∩ B ) ∪ C = ( A ∪ C ) ∩ ( B ∪ C )
- De Morgan law.
Access to our continued use python to explain.
Here we use python to achieve operational collection:
A = set([1, 2, 3, 4,5,6]) B = set([3, 4, 5, 6,7,8]) print (A & B) # intersection intersection print (A | B) # set union and print (A - B) # differencing sets (items A, B, but not in) difference, element in A, and not in B print (A ^ B) # symmetric difference (items in A or B but not both occur simultaneously in the symmetric difference, (A | B) - (A & B)
Here we use to determine whether the element belongs in the collection, as well as with>,> =, <, <= determines the belongingness of the two sets, such a set is a subset of another set.
A = set([1, 2,3,]) B = set([1, 2,3,4,5,6]) print(1 in A) # element print(A < B) # subset
Here we return the total number of elements in the collection, a collection of maximum, minimum collections, add and delete elements Note Depending on probability theory, the collection will not have duplicate elements:
A = set([1, 2,3,4,5,6]) set_len = len (A) print(set_len) A.add(6) # add an element print(A) A.remove(2) # remove an element print(A) A.add(1) print(A) # a set has no repeated elements
When inconvenient to use python, unfortunately Microsoft does not support the set operation math app, here is the Microsoft mathematics supported by mathematical calculations .
Useful links:
Probability theory (a) random events and probability
The relationship between random events and events
Probability theory - the first chapter] [random events and probability
Math vocabulary (Probability and Statistics)
Mathematics English vocabulary Daquan
Linear Algebra large collection of English vocabulary
Common words in bilingual mathematics
Python collection (set) type of operation
To facilitate the search for information, now list the English term term used in this section:
- Collection - set
- Randomized trial - Random experiment
- Sample space - sample space generally expressed Ω
- Sample points - to be confirmed
- Random Events - Random event
- Basic events - Elementary event
- Unlikely event - Impossible event
- Random event - to be confirmed
- Complete Event Group - to be confirmed
- A⊂B - to be confirmed
- A = B - to be confirmed
- A∪B - to be confirmed
- A∩B - to be confirmed
- Commutative law - to be confirmed
- Associative law - to be confirmed
- Distributive law - to be confirmed
- De Morgan law - to be confirmed