Mutually exclusive is mutually exclusive, and mutually exclusive is not independent.
For independent events A and B, there is P(AB)=P(A)*P(B).
For mutually exclusive events, P(AB)=0.
Is there any knowledge of probability theory in mathematical analysis? No, but in solving the problem of probability theory, we sometimes need to use the knowledge of mathematical analysis, such as integration.
Proved by the knowledge of probability theory, if the rule of the lottery is that everyone draws and then puts it back for the next person to draw, it is an average problem. Each draw has nothing to do with the previous results, and its probability is 1/n (similar to flipping a coin).
If you don't put it back after smoking, the result will be different. The probability at this time is related to the previous results.
The probability that the first person draws is 1/n+ 1/(n-2)+ ...
The probability that the second person draws is1/(n-1)+1/(n-3)+ ...
This is related to the value of n.
For the simplest example, when n= 1, the probability of the first person drawing is 1, and the probability of the second person drawing is 0.
Will the calculation method use the knowledge of probability theory? No, it will use knowledge of algebra and analysis.
The knowledge of probability theory that A and B do not contain is that A and B are independent of each other. Incompatibility means that it does not happen at the same time, that is to say, A and B still influence each other. Independence is a, and it doesn't matter whether B happens or not.
Given that P(A)=0.7 and P(A-B)=0.3, try to find the knowledge solution of P (the antonym of AB) probability theory, and P (a-b) = P (a)-P(AB).
P(AB)=0.7-0.3=0.4
P (the antonym of AB) =1-P(AB) =1-0.4 = 0.6.
The solution to the problem of probability theory: suppose there are the number of balls in the first cup (the same is true for other cups)
Let x be a random variable of the number of balls in the first cup,
The distribution law of the number of balls in the first cup;
X 0 1 2 3
Probability 27/64 27/64 9/64 1/64
Try to use the knowledge of probability theory to explain that you can't put all your eggs in one basket when buying lottery tickets. Let me give you an example:
Suppose 1 10,000 lottery tickets are issued, and each 5 yuan has 5 first prizes, 3 1.5 million yuan, 95 second prizes, 5,000 yuan, 900 third prizes, 300 yuan prizes, 9,000 fourth prizes and 20 yuan prizes.
Then the expected income from spending five dollars on lottery tickets is: 315000× 5/100000+5000× 95/1000000+300× 900/100000+20× 900.
This is obviously far less than 5 yuan.
Put all your eggs in one basket to buy lottery tickets, which means spending a lot of money on lottery tickets. Suppose you bought n tickets.
Then buy lottery tickets as an independent event, recorded as X 1.
Then the mathematical expectation of X 1+X2+ ... +Xn is 2.5n
Obviously, the more money you invest, the more money you may lose from the expected income.
What do you want to ask about the knowledge of relativity? Just add a few words to answer.
Will the knowledge of complex analysis be used in advanced probability theory? Basically not. Most of the content is the knowledge of measurement theory.
But when it comes to the knowledge of characteristic function, we need to use some knowledge of complex variable function, and we just use residue theorem to calculate some integrals. However, the content of characteristic function is very important, which is related to convergence according to distribution and the central limit theorem below.
To sum up, as long as you know the knowledge of residue calculation integral, you don't know more about complex variable function. But the residue theorem itself involves a lot of basic knowledge of complex analysis.