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Probability that an event that can only occur once will occur? 
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#1
Jun1512, 08:15 PM

P: 300

According to the FBI, the probability of being murdered within a year is 1/18690. Assuming that a person is alive at age 12 and will die at 82 due to natural causes if that person isn't shot before then, what is the probability that he will be murdered?
Obviously, whether or not the person is alive at a certain time between age 12 and 82 would depend on whether or not he or she was murdered before then. Therefore, a murder can only happen once. 


#2
Jun1512, 09:52 PM

P: 6

Alright, here is the solution. Take 60 points to consider. It is a summation.
Event No.1Probability of being murdered in first year(starting from 12). is (c = 1/18690). Event No.2Probablity of being not murdered in 1st year AND being murdered in 2nd year = (1c)*c. Event No.3Probability of no murder in 1st and 2nd year but in 3rd = (1c)*(1c)*c and so on....... till 60.. You have to add them all. You fill find that it is a geometric progression. Take 'c' common to get something like P = c(1+(1c)+(1c)^2 +(1c)^3......till (1c)^59)... Add them using the summation rule for geometric progression, if u don't know, Google is your friend. The answer that you'll get is....3.2*10^3. thanks..it was a good question. P.S. In fractions, it is 2/625. The probability increases with age, yes. 


#3
Jun1612, 12:00 AM

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Simplest to approximate as a negative exponential: prob of not being murdered in next 70 years is (11/18690)^{70} ≈ e^{70/18690} = 1  0.0037.
So prob of being murdered is about 0.37%. cheekujodhpur, what did you mean by "the probability increases with age"? If you increase the 82, sure, but if you take 82 as fixed and ask, for your current age, what is the probability that you will die violently then it decreases with age (as you'd expect). 


#4
Jun1612, 06:16 AM

P: 6

Probability that an event that can only occur once will occur?
"If you increase the 82, sure, but if you take 82 as fixed and ask, for your current age, what is the probability that you will die violently then it decreases with age (as you'd expect)."
Yes, that's exactly what I meant. I got 0.32%, so that's the same. I just took 60 instead of 70, my mistake! But I don't know much about the approximation. Can you give a link or just post why the approximation is valid. Is it for probabilities or for any geometric progression? 


#5
Jun1612, 06:27 AM

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#6
Jun1612, 06:33 AM

P: 6

Thanks!



#7
Jun1612, 06:40 AM

Mentor
P: 15,066

Or just use the binomial expansion and bypass the exponential function, (1x)^{n} = 1nx+O(x^{2})



#8
Jun1612, 12:32 PM

P: 300

Hey, thanks for your help guys. For a minute there, I thought that this theoretical person could not safely expect to live to be 82 years old.



#9
Jun1612, 01:04 PM

P: 2,499

In terms of the probability of being murdered, the model would not hold for the 100,000 year old person. In terms of the model, probably the best one can do is assume the proportion of causes of death would be constant. The calculation above needs to be corrected for overall survival in terms of death from any cause. 


#10
Jun1612, 01:05 PM

P: 250

If you consider that the number of people we know plus acquaintances can easily be around 300 persons that would mean that most 82 year old persons know of someone in their circles who has been murdered. Mmm... that might be an interesting survey. 


#11
Jun1612, 01:41 PM

P: 2,499




#12
Jun1612, 01:50 PM

P: 300

This person will not die until he reaches age 82, if he is not murdered. As mentioned before, this person has a 68% chance of living to be 82. 


#13
Jun1612, 02:02 PM

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P: 15,066




#14
Jun1612, 02:40 PM

P: 2,499

If you're asking the probability of being murdered given you are otherwise immortal, you will have a ridiculously long mean life expectancy. This is why there are so many jokes about statistics. 


#15
Jun1612, 02:56 PM

P: 250




#16
Jun1612, 06:00 PM

P: 2,499

The mistake is trying to make murders independent of all deaths Being murdered is conditional on being dead although the reverse is not true. 


#17
Jun1612, 06:47 PM

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P: 15,066

This is a basic probability problem whose goal is to show students how to calculate simple probabilities. It is nothing more than that. You are reading far too much into this simple problem. Suppose the problem had been written as According to the Froboz Company, one out of 18690 Froboz widgets fail every month. Assuming such a widget is used in a device that will be dismantled in 70 months, what is the probability that the widget will fail before the device is dismantled?The answer is exactly the same, about 0.003738, or 0.3738%. Just as the original wording ignores finer details such as the fact that the murder rate varies with age, this rewording also ignore finer details such as the bathtub curve nature of device failures. Those finer details are irrelevant and they are a derailment of this thread. It's obvious that this is a simplification as there are lots of ways people can die besides being murdered and dropping dead of old age at 82. It's obvious that this is a simplification because the murder rate depends on a lot of factors: Age, where one lives, income bracket, whether or not one's idea of a good time is to go pick a fight at the neighborhood biker bar, ... Because of this derailment over whether this question is valid, nobody has pointed out that there are two ways to solve this problem:
The former method is rather nasty while the latter is very simple. This should be the takeaway point from this question. 


#18
Jun1612, 07:02 PM

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