Are both of these the same thing? (logical NOT and the complement rule)

Hi everyone,

This is an example of binary variable called as logical NOT https://www.fico.com/fico-xpress-op.../mipform/dhtml/chap2s1.html?scroll=sseclognot

...and this is the complement rule of probability https://brilliant.org/wiki/probabil...tal to 1.&text=Equivalently, P ( A c ),1−P(A)

Are both of these things the same?

The logical NOT is the same concept as the complement in set theory.

PeroK said:

The logical NOT is the same concept as the complement in set theory.

Thank you.

A logical variable is a True/False or a 1/0 variable. Probabilities can have any real value from 0 to 1, including 0 and 1. In other words, it can express True/False/Maybe. In a coin toss, the event of Tails is the complement of the event of Heads. Both have a probability of 1/2.

Thank you for these responses. I wonder if we can use probability formulas instead of crime per capita (for example in this context, rape) in order to see the likelihood of which of these two US states are more likely to commit rape against someone. For instance, Alaska has a rape rate of 134.0 per 100k and Tennessee has a rape rate of 38.2 per 100k according to https://en.m.wikipedia.org/wiki/List_of_U.S._states_and_territories_by_violent_crime_rate. How do we formulate a probability formula question using these datas?

wonderingchicken said:

For instance, Alaska has a rape rate of 134.0 per 100k and Tennessee has a rape rate of 38.2 per 100k according to https://en.m.wikipedia.org/wiki/List_of_U.S._states_and_territories_by_violent_crime_rate. How do we formulate a probability formula question using these datas?

Suppose R is the event of a particular person being raped, A is the event that the person is in Alaska, and T is the event that the person is in Tennessee. Then the probability of a particular person being raped, given he is in Alaska is denoted P(R|A) = 134/100,000 = 0.00134. The probability of a particular person being raped, given he is in Tennessee is denoted P(R|T) = 38.2/100,000 = 0.000382.

FactChecker said:

Suppose R is the event of a particular person being raped, A is the event that the person is in Alaska, and T is the event that the person is in Tennessee. Then the probability of a particular person being raped, given he is in Alaska is denoted P(R|A) = 134/100,000 = 0.00134. The probability of a particular person being raped, given he is in Tennessee is denoted P(R|T) = 38.2/100,000 = 0.000382.

Thank you so much. But in the real world, per capita tells us more details than using probability formula. Is that accurate? Sorry.

wonderingchicken said:

Thank you so much. But in the real world, per capita tells us more details than using probability formula. Is that accurate? Sorry.

I can't think of anything else it says unless some more facts are given.

FactChecker said:

I can't think of anything else it says unless some more facts are given.

Alright. Let's see if I can formulate a probability question by myself correctly.

Say, the numbers of crime in population A which is around 15-16 millions (let's say 15 millions) are 889 crimes vs the numbers of crime in population B which is around 106-107 millions (106 millions) are 3333. As we understand, probability formula is favorable event divided by numbers of possible events. 889 divided by 15 millions is 0.00005 and 3333 divided by 106 millions is 0.00003.

But because there are too many zeroes in there, can we do this instead? What I mean is, for example for population A we just divide 889 by 1500 so we will get 0.5 instead of 0.00005 when we divide 889 by 15 millions. It is still centered around 5 but with less zeroes. Also, the probability range is from 0 to 1.

wonderingchicken said:

Say, the numbers of crime in population A which is around 15-16 millions (let's say 15 millions) are 889 crimes vs the numbers of crime in population B which is around 106-107 millions (106 millions) are 3333. As we understand, probability formula is favorable event divided by numbers of possible events. 889 divided by 15 millions is 0.00005 and 3333 divided by 106 millions is 0.00003.

But because there are too many zeroes in there, can we do this instead? What I mean is, for example for population A we just divide 889 by 1500 so we will get 0.5 instead of 0.00005 when we divide 889 by 15 millions. It is still centered around 5 but with less zeroes. Also, the probability range is from 0 to 1.

I see your point that the numbers are very small for this example. Probabilities are just another way of representing the same facts as the rate per 100,000. But it is valid to compare probabilities (or rates). You can say that the probability of being raped in Alaska is 3.8 times higher than in Tennessee. (0.00134/0.000362 = 134/38.2 = 3.70)

FactChecker said:

I see your point that the numbers are very small for this example. Probabilities are just another way of representing the same facts as the rate per 100,000. But it is valid to compare probabilities (or rates). You can say that the probability of being raped in Alaska is 3.8 times higher than in Tennessee. (0.00134/0.000362 = 134/38.2 = 3.70)

Yes, even from your previous calculations for Alaska it is 0.00134 and for Tennessee it is 0.000382. The probability range for Alaska is higher than Tennessee, therefore the probability of being raped is higher in Alaska.

But for my own formulated probability question, is it appropriate if I change the population from 15 millions to 1500 to get appropriate numbers so we can use the probability range which is from 0 to 1? By the way, in my formulated probability question instead of using rate I was using the numbers of crime or criminals.

wonderingchicken said:

Alright. Let's see if I can formulate a probability question by myself correctly.

Say, the numbers of crime in population A which is around 15-16 millions (let's say 15 millions) are 889 crimes vs the numbers of crime in population B which is around 106-107 millions (106 millions) are 3333. As we understand, probability formula is favorable event divided by numbers of possible events.

That is valid if all the events have an equal probability. Otherwise, it is not valid.

wonderingchicken said:

889 divided by 15 millions is 0.00005 and 3333 divided by 106 millions is 0.00003.

But because there are too many zeroes in there, can we do this instead? What I mean is, for example for population A we just divide 889 by 1500 so we will get 0.5 instead of 0.00005 when we divide 889 by 15 millions. It is still centered around 5 but with less zeroes. Also, the probability range is from 0 to 1.

It does not represent a probability if you divide by the wrong number. That is important. You do not want to tell a person that he has a 50% probability of being a victim of crime when the reality is that he has a 0.005% probability of being a victim.

FactChecker said:

That is valid if all the events have an equal probability. Otherwise, it is not valid.

It does not represent a probability if you divide by the wrong number. That is important. You do not want to tell a person that he has a 50% probability of being a victim of crime when the reality is that he has a 0.005% probability of being a victim.

Ok. Thanks for the important informations. For population A which is 889 crimes or criminals divided by 15 millions is 0.00005 and for population B which is 3333 crimes or criminals divided by 106 millions is 0.00003.

Therefore, from probability range 0 to 1, the probability of someone getting assaulted by a criminal in population A is 0.00005 and the probability of someone getting assaulted by criminal in population B is 0.00003. So, someone getting assaulted by a criminal is more likely in population A than population B.

Is this correct?

wonderingchicken said:

probability of someone getting assaulted

The answer is yes you are correct . Of course the semantics matter very much. This is the probabiliity of assault on some particular person (someone). The probability of "anyone" being assaulted will depend explicitly upon the population. Most fallacious inferences (I like to call them call them "Trumpisms") depend upon this type of error.

hutchphd said:

The answer is yes you are correct . Of course the semantics matter very much. This is the probabiliity of assault on some particular person (someone). The probability of "anyone" being assaulted will depend explicitly upon the population. Most fallacious inferences (I like to call them call them "Trumpisms") depend upon this type of error.

Thank you. So, the favorable event can be rate, numbers of everything, anything, etc.?

So long as they are correctly used (usually this is a question of "normalizing " the compared ratios in a fashion that makes sense.) ##\frac 3 5 \neq \frac 3 8## even though ##3=3##

hutchphd said:

So long as they are correctly used (usually this is a question of "normalizing " the compared ratios in a fashion that makes sense.) ##\frac 3 5 \neq \frac 3 8## even though ##3=3##

Sorry for not telling the details. I mean both the favorable event or outcome and the total events or outcomes can be expressed as the numbers of crimes committed or numbers of criminals (or even rate per 100k, and so on) as the favorable outcome and the total population of something as the total outcome, isn't it?

Just like, for example, 10 red marbles as the favorable outcomes divided by 20 marbles as the total outcomes.

I keep forgetting about probability formula because sometimes it messes up my thinking trying to make sense both the question and the formula.