Problem of Calculating Probabilities

In summary, the conversation is about calculating the probability of a number N being a product of a prime below its square root. The probability depends on whether N is a product of 2, 3, or 5, and there is a discussion about how to arrive at a combined probability for all primes up to sqrt N. Various methods are mentioned, such as sieving and using the sum of reciprocals of primes, but there are challenges in accounting for irrelevant primes.
  • #1
PeterJ1
17
0
I hope this question is in the right place.

I'm trying to calculate probabilities and struggling. Hopefully someone can help.

Suppose I want to calculate the probability of N being a product of a prime below sqrt N.

N will have a 1:2 chance of being a product of 2.

If N is not a product of 2 then it will have a 1:6 chance of being a product of 3.

If N is not a product of 2,3 then it will have a 1:15 chance of being a product of 5.

And so on...

How would I arrive at combined probability for the primes up to sqrt N?

It could be a different calc. I could sieve out the products of each primes at each step and then calculate the probabilities for only the numbers that remain. So, for the divisor 2 the odds are 1:2. For 3 the odds are 1/3 but half the numbers have already been eliminated so really it is 1/6. Then I could use 1/3p as a simple estimate for each successive potential divisor, but would have no idea how to sum them.

Either way I cannot see an easy way to do it.

(I know the the PNT gives the probability of a number being prime, by the way, but this is not where I'm going.)

Thanks for any help with this. No doubt the solution looks easy for you guys.

Another way to do it seems to be use a table for the sum of the reciprocals of primes up to sqrt N, but I want to be able to deduct from this primes that are irrelevant. So, if I multiply 2,3,5, 7 and deduct 1 to give N, then I know these primes can be ignored. If I deduct the odds of these four primes being divisors then I can deduct this from the sum of the reciprocals up to N and get some sort of result. But if there are 100 possible divisors to deduct then I'm back where I started, trying to work out the sum of probabilities for these 100 possible divisors.

If I've asked a **** fool question then my apologies.
 
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  • #2
Re: Problem of Calculating Probabilitiers

PeterJ said:
I hope this question is in the right place.

I'm trying to calculate probabilities and struggling. Hopefully someone can help.

Suppose I want to calculate the probability of N being a product of a prime below sqrt N.
Do you mean "having a prime factor below sqrt(N)"?

N will have a 1:2 chance of being a product of 2.
Do you mean "being a product of two primes" or do you mean "having 2 as prime factor"? In either case, how do you arrive at "a 1:2 chance"?

If N is not a product of 2 then it will have a 1:6 chance of being a product of 3.

If N is not a product of 2,3 then it will have a 1:15 chance of being a product of 5.

And so on...

How would I arrive at combined probability for the primes up to sqrt N?

It could be a different calc. I could sieve out the products of each primes at each step and then calculate the probabilities for only the numbers that remain. So, for the divisor 2 the odds are 1:2. For 3 the odds are 1/3 but half the numbers have already been eliminated so really it is 1/6. Then I could use 1/3p as a simple estimate for each successive potential divisor, but would have no idea how to sum them.

Either way I cannot see an easy way to do it.

(I know the the PNT gives the probability of a number being prime, by the way, but this is not where I'm going.)

Thanks for any help with this. No doubt the solution looks easy for you guys.

Another way to do it seems to be use a table for the sum of the reciprocals of primes up to sqrt N, but I want to be able to deduct from this primes that are irrelevant. So, if I multiply 2,3,5, 7 and deduct 1 to give N, then I know these primes can be ignored. If I deduct the odds of these four primes being divisors then I can deduct this from the sum of the reciprocals up to N and get some sort of result. But if there are 100 possible divisors to deduct then I'm back where I started, trying to work out the sum of probabilities for these 100 possible divisors.

If I've asked a **** fool question then my apologies.
 
  • #3
Re: Problem of Calculating Probabilitiers

HallsofIvy said:
Do you mean "having a prime factor below sqrt(N)"?

Yep.

Do you mean "being a product of two primes" or do you mean "having 2 as prime factor"?

Having 2 as a factor.

In either case, how do you arrive at "a 1:2 chance"?

Er, one in every two numbers is divisible by 2.
 

FAQ: Problem of Calculating Probabilities

What is the problem of calculating probabilities?

The problem of calculating probabilities is a fundamental issue in mathematics and statistics. It refers to the challenge of determining the likelihood of a particular event or outcome occurring, based on available information and data.

Why is calculating probabilities important?

Calculating probabilities is important because it allows us to make informed decisions and predictions based on logical reasoning and evidence. It is especially useful in fields such as science, finance, and gambling, where understanding and quantifying uncertainty is crucial.

What are some common methods for calculating probabilities?

There are several methods for calculating probabilities, including the classical method, relative frequency method, and subjective method. The classical method involves using theoretical probabilities based on equally likely outcomes, while the relative frequency method uses observed data to estimate probabilities. The subjective method, on the other hand, relies on personal judgment and beliefs to assign probabilities.

What are some challenges in calculating probabilities?

One of the main challenges in calculating probabilities is the lack of complete information. Often, we do not have all the data needed to accurately determine the likelihood of an event. Additionally, the complexity of the problem and the assumptions made can also affect the accuracy of the calculated probabilities.

How can we improve the accuracy of calculated probabilities?

To improve the accuracy of calculated probabilities, we can collect more data, use more advanced statistical techniques, and carefully consider the assumptions and limitations of the problem. It is also helpful to seek feedback and input from other experts in the field to ensure a more comprehensive and accurate analysis.

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