Solve Probability Inequality | Finite Probabilistic Space

In summary, the conversation discusses the definition of a finite probabilistic space and two events A and B with unequal probabilities. It is then shown that by choosing a specific probability distribution on the sample space, the inequality Pr[A∩B]>=9*Pr[A]*Pr[B] can be satisfied for the events A and B.
  • #1
alfred2
10
0
Hi! I have to do this exercise:

Define a finite probabilistic Space (Ω; Pr[ ]) and 2 events A,B⊆ Ω and Pr[A] ≠ Pr so that we can verify that
Pr[A∩B]>=9*Pr[A]*Pr > 0. (1)
___________________________________________
I've been trying it but i have reached this conclusion:
If Pr[A]>0 Pr[A]=Pr[A∩B]/P[B|A]
IF Pr>0 Pr=Pr[A∩B]/P[A|B]
Substituting in (1) we have:
P[A|B]*P[A|B]>=9*Pr[A∩B]
I don't know if this help. Can anyone help me please? Thanks
 
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  • #2
I solve the above question from an unanswered MHB thread.

Consider $\Omega=\{a_1,a_2,\ldots,a_m\}$ and the probabilty on $\mathcal{P}(\Omega)$ defined by: $$p(a_i)=\frac{1}{m}\quad \forall i=1,2,\ldots,m$$ Consider $A=\{a_1,a_2\}$ and $B=\{a_1,a_3\}$. Then, $$p(A\cap B)\ge 9p(A)p(B)\Leftrightarrow \frac{1}{m}\geq 9\cdot \frac{2}{m}\cdot\frac{2}{m}\Leftrightarrow 1\ge \frac{36}{m}\Leftrightarrow m\ge 36$$ That is, you only need to choose $\Omega$ with 36 elements.
 

FAQ: Solve Probability Inequality | Finite Probabilistic Space

What is a finite probabilistic space?

A finite probabilistic space is a mathematical concept used to model situations where there are a limited number of possible outcomes and each outcome has a known probability of occurring. It is often represented as a sample space, which lists all the possible outcomes, and a probability function, which assigns a probability to each outcome.

How do you solve probability inequalities?

To solve a probability inequality, you first need to determine the probability of the event or outcome you are interested in. This can be done by using the probability function and applying any relevant probability rules or formulas. Then, you can use algebraic methods to manipulate the inequality and solve for the unknown variable.

What is the difference between probability and probability inequality?

Probability refers to the likelihood of a specific event or outcome occurring, while probability inequality involves comparing the probabilities of two different events or outcomes. Probability inequality is often used to determine which event is more likely to occur or to establish a minimum or maximum probability for a given event.

What are some common examples of finite probabilistic spaces?

Examples of finite probabilistic spaces include rolling a die, flipping a coin, drawing cards from a deck, and playing games like roulette or bingo. These situations have a limited number of possible outcomes and each outcome has a known probability of occurring, making them ideal for modeling with a finite probabilistic space.

How is finite probabilistic space used in real life?

Finite probabilistic space is used in a variety of fields, including finance, statistics, and biology. In finance, it is used to model investment strategies and risk management. In statistics, it is used to analyze data and make predictions. In biology, it is used to model genetic inheritance and population dynamics. It is also used in everyday decision making, such as determining the likelihood of winning a game or making a profit on a purchase.

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