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If the average lifetime of a person is $72.6$ and if the person has a $0.00016\%$ chance of being killed by doing a certain activity in a year.

How do you calculate the chance of him being killed doing that activity in a lifetime?

Is it $(72.6 * 0.00016)=0.011616\% ?$ (It might sound simple but If possible, please provide a clue/explanation on how to calculate such problems.)

Or Shahar
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user107785
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  • Well, one problem with that is it allows for dying from that activity several times over the course of your life. Of course, you could argue that, as the event is so unlikely, the probability that it happens several times is negligible. A more serious problem is that it assumes independence, which is highly unlikely. If the event is, say, sky diving then one reason the probability is so low is that most people never sky dive. – lulu Jan 01 '21 at 15:48
  • @lulu Thanks for the reply. I'm not able to grasp the "dying from that activity several times over the course of your life".

    As for the second portion, Aye! most people never skydive so their chance of dying due to it will be less but if we were to go under the assumption that everyone has experience in skydiving will the way I'm calculating be right?

     – user107785 Jan 01 '21 at 16:05
  • No, dependence is still a big problem. For the first issue, to die in year $n$ of the event, you must have not died in years $1$ through $n-1$ (of any causes). You can't just add this way. If you had a $2%$ chance of dying from some particular thing in a given year, your method would give you a $145%$ chance of dying from it over your life, which is absurd. – lulu Jan 01 '21 at 16:14
  • The point of the dependence is that there is nowhere near enough information provided for us to answer the question. Suppose the event is "complications from pregnancy". Then well over half the population is sure never to die from this. Even those who are susceptible to this are only susceptible a few times over their lives. In practice, one would want to consult real data to determine the effective probability. – lulu Jan 01 '21 at 16:22
  • @lulu Damn your right! that is indeed absurd. Thank you very much for the explanation as well, it was very easy for me to understand.

    I was basically doing a bit of reading on Police Shootings and was trying to figure out the math. I saw that in a year 405 white people were killed by shootings and came to a conclusion that in a given year {405 / 250.4Million<-pop of white people *100 = 0.00016%approx are killed} and was trying to find the % chance a white person can be shot across their lifetime and that's where I ended up reaching a dead end.

     – user107785 Jan 01 '21 at 18:18

1 Answers1

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If the probability of a single person experiencing a certain condition (such as being killed, in your example) each year is $p$ then we can say that the probability of a single person not experiencing that condition in a year is $1-p$.

Then, the probability of a person never experiencing that condition throughout their $n$ years of life can be calculated as the product of probabilities of not experiencing that condition in every single one of those $n$ years. In your example, if a person is to not ever get killed in their life, that person should not be killed in their year 1, and not in year 2, and not in year 3, etc. So, the probability of not experiencing that certain condition in $n$ years is $(1-p)^n$ .

Now, the probability of experiencing that condition (in your example, getting shot) will be: $$1 - (1-p)^n$$

In your example, $p=\frac{405}{250.4million}=0.0000016$ and $n=72.6$ , so: $$1 - (1-p)^n = 0.000117$$ In other words, in your example a person has a nearly $0.01$ percent chance of experiencing that particular condition in their life.

Saeed
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  • Thank you so much for your answer! I had to read and re-read it a couple of times but finally got a hang of it. But I have a doubt, by always having n=72.6, aren't we assuming that each person starts at age 1 if we were to use this method?

    There might be many people who are already say 50years during a given year and having n as 72.6 for all the cases might not be right here? (I'm going to read more into such problems, I believe you used Binomial distribution. I'll start from there)

     – user107785 Jan 02 '21 at 07:39
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    You are right: this solution is based on binomial distribution. As for $n$ , I have used an approximation here and allowed myself to use 72.6 as exponent, whereas it one can reasonably argue that $n$ is the number of experiments and has to be a natural number. Finally, regarding your question, your guess is right: we do not account for years that people have survived the fatal circumstance so far. Our calculated probability is for each person at the beginning of their life. If we wish to calculate the probability of survival of a 50-year old in the rest of their life, we can use $n=72.6-50$ – Saeed Jan 02 '21 at 07:55