Show A_k={X is divisible by p_k} independent events

In summary, The conversation is discussing the concept of prime numbers and how they relate to the Fundamental Theorem of Arithmetic (FTA). The idea is that all natural numbers can be represented as a unique product of primes. The goal is to show that certain events, involving integers that are the product of remaining primes, are independent. However, there may be some confusion about how to handle the implied union inside the brackets.
  • #1
Gregg
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Homework Statement



##p_k## refers to prime numbers ##p_1<\cdots<p_k<\cdots ##

I thought that it could have something to do with the FTA so all natural numbers are composed of a unique product of primes.

So I need to show ## \text{P} ( X=M p_l \cap X = N p_m ) = \text{P} ( X=M p_l ) \text{P}( X = N p_m ) ## and this shows that all of the events are independent doesn't it...

M,N are integers which are the product of the remaining primes. So actually, there is an implied union inside those brackets which means i have to take the union of all integers?
 
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  • #2
I'm not sure how to proceed and I'm a bit confused. Any help would be appreciated, thanks!Homework Equations No equations needed, just need help with the logic of it. The Attempt at a Solution See above
 

FAQ: Show A_k={X is divisible by p_k} independent events

What is the definition of independent events?

Independent events are events that have no influence on each other. This means that the outcome of one event does not affect the outcome of the other event.

How do you determine if two events are independent?

To determine if two events are independent, you can use the multiplication rule. If the probability of both events occurring together is equal to the product of their individual probabilities, then the events are independent.

What is the formula for calculating the probability of independent events?

The formula for calculating the probability of independent events is P(A and B) = P(A) * P(B). This means that the probability of both events occurring together is equal to the product of their individual probabilities.

How does the concept of independence apply to event A_k={X is divisible by p_k}?

The concept of independence applies to event A_k={X is divisible by p_k} in that the likelihood of X being divisible by p_k is not affected by the values of other events. For example, if X is divisible by 2, this does not affect the probability of X being divisible by 3.

Can independent events occur simultaneously?

Yes, independent events can occur simultaneously. This means that both events can happen at the same time without affecting each other's probability of occurring.

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