How Do Probability Properties Influence Event Outcomes?

In summary, probabilities are a measure of the likelihood of an event occurring. There are three main types of probabilities: theoretical, experimental, and subjective. These probabilities are calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Probability and statistics are closely related fields, with probability used for predictions and statistics used for data analysis. In science, probabilities are important for making predictions, understanding uncertainty, and testing hypotheses.
  • #1
Kudasai
4
0
Let $(\Omega,\mathcal F,P)$ be a space of probabilities and $(A_n)_n\subseteq \mathcal F.$ Show that

a) if the sequence satisfies $\inf\{P(A_n)_n:n\ge1\}=\alpha,$ with $\alpha\ge0,$ then $P\left( \bigcap\limits_{n=1}^{\infty }{\bigcup\limits_{k=n}^{\infty }{{{A}_{k}}}} \right)\ge \alpha .$

b) if the events $A_1,\ldots,A_n$ satisfy $P\left( \bigcap\limits_{i=1}^{n}{{{A}_{i}}} \right)=1$ then $A_1,\ldots,A_n$ are independent events.

c) If $A,B\in\mathcal F$ then show that $P(A\cap B)\ge P(A)+P(B)-1$

Any ideas to solve a)? For b), I should prove that $P(A_1\cap\cdots\cap A_n)=P(A_1)\cdots P(A_n),$ right? But I don't see how to use the fact I'm given.
As for c), I think I'm missing some property of sets, or any trick there. How to takle it?

Thank you.
 
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  • #2
a) By the Monotone Convergence Theorem, we have$$P\left( \bigcap\limits_{n=1}^{\infty }{\bigcup\limits_{k=n}^{\infty }{{{A}_{k}}}} \right)=\lim_{n\to\infty}\,P\left(\bigcup_{k=n}^{\infty }{{{A}_{k}}}\right)=\lim_{n\to\infty}\,\sum_{k=n}^{\infty}P(A_k)\ge \lim_{n\to\infty}\,\inf_{k\ge n}\,P(A_k)=\alpha.$$b) Let $A_1,\ldots,A_n\in\mathcal F$ such that $P\left( \bigcap\limits_{i=1}^{n}{{{A}_{i}}} \right)=1.$ We have$$1=P\left( \bigcap\limits_{i=1}^{n}{{{A}_{i}}} \right)=P(A_1\cap\cdots\cap A_n)=P(A_1)\cdots P(A_n).$$This shows that $A_1,\ldots,A_n$ are independent events.c) Since $A,B\in\mathcal F$, we have$$P(A\cap B)\ge P(A)+P(B)-1$$by the subadditivity of the probability measure.
 

FAQ: How Do Probability Properties Influence Event Outcomes?

What are probabilities?

Probabilities are a measure of the likelihood of an event occurring.

What are the different types of probabilities?

There are three main types of probabilities: theoretical, experimental, and subjective. Theoretical probabilities are based on mathematical calculations and assumptions. Experimental probabilities are based on actual data and observations. Subjective probabilities are based on personal beliefs or opinions.

How are probabilities calculated?

Probabilities are calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This can be represented as a fraction, decimal, or percentage.

What is the relationship between probability and statistics?

Probability and statistics are closely related fields. Probability is used to predict the likelihood of an event occurring, while statistics is used to analyze and interpret data to make inferences about a population or sample.

Why are probabilities important in science?

Probabilities are important in science because they allow us to make predictions and decisions based on data and evidence. They also help us understand the uncertainty and variability in natural phenomena, and can be used to test hypotheses and theories.

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