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I am hoping that this is not too basic a question for this site, but I am seeking to better understand a conversation I have had with my more mathematically inclined friend.

The discussion was around the probability of an event occuring with repeated exposure to a specific activity. The example in the conversation was the risk of (for example) injury occuring compared between someone who was a frequent skydiver (perhaps weekly) and someone who skydives once or twice a year.

For the point of this exercise we are ignoring all external factors (wind, experience, etc) and are assuming that the risk of any injury (i.e. not one that would preclude someone from continuing to skydive) is fixed for each event. That is, as with a coin toss, each individual event is unaffected by the other events.

To my mind it makes more sense that the person who chooses to skydive regularly is accepting a higher risk over the course of the year then the person who skydives once a year. My friend contests that the risk of an event happening is equal for both people over the course of the year. He attempted to explain this to me, but could not in a way that I understood.

To clarify, I don't believe the risk increases cumulatively (i.e. if there was a 1/10 chance of something happening in one dive then I understand that this does not mean that something will definitely happen if a person dives 10 times).

I am quite curious about which answer is correct, probability was always one of those areas that I sometimes found to be non-intuitive. Any insight here would be appreciated

Thanks!

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    $\begingroup$ Does your friend also think if you roll 1000 dice, the probability to hit atleast one six is the same as if you only rolled one dice? That must be some magical dice. If I am firing at you with a gun, do you think you have the same chance to survive if you stay in my line of sight instead of hiding? No, that answers your question $\endgroup$
    – TROLLHUNTER
    Mar 30, 2011 at 15:34
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    $\begingroup$ Based upon your description of his arguments, I wish your friend was much more mathematically inclined than he is... $\endgroup$
    – Pete L. Clark
    Mar 30, 2011 at 15:42
  • $\begingroup$ hehe, thanks @solomoan, good point ;) $\endgroup$
    – Chris
    Mar 30, 2011 at 15:49
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    $\begingroup$ It is interesting to me how often this question comes up in different contexts. $\endgroup$
    – rcollyer
    Mar 30, 2011 at 15:55
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    $\begingroup$ Coming up next: The risk of having a bomb on board of an airplane is $\frac1{1000}$. So to be safe, take a bomb with you, cause the probability of two bombs is just $\frac1{1000000}$. $\endgroup$
    – Hagen von Eitzen
    May 11, 2013 at 21:04

3 Answers 3

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Suppose the chances you die from a skydive are 0.1.

If you repeat this 10 times, the chances you die from any one of the dives is

$$1 - (0.9)^{10} = 0.6513...$$

So even though the probability of dying on a specific dive does not change, by repeatedly skydiving you are increasing your overall chances of dying.

So if someone says they will skydive once in 2011, the chances that they are alive (not counting other factors) in 2012 is 90%, but if someone else dives 10 times, the chances they are alive in 2012 is just 35%.

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    $\begingroup$ The above formula works even if you assume he stops skydiving after he dies :-) $\endgroup$
    – Aryabhata
    Mar 30, 2011 at 15:46
  • $\begingroup$ Fantastic, thanks for that :) $\endgroup$
    – Chris
    Mar 30, 2011 at 15:48
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Moron's example nicely illustrates the importance of time to issues of risk.

Consider the recent pandemonium about radiation exposure in the Fukushima Dai-ichi nuclear power plant.

Any discussion about the dangers of radiation needs to be put in the context of amount of exposure. "Readings" in milliSieverts per hour only inform us about the rate of radiation leakage. Accumulated risk corresponds to milliSieverts; it is a rate x time. It's a little like the distinction between speed and distance. Accumulated risk is more like distance.

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  • $\begingroup$ The effect of radiation exposure is not necessarily sublinear. $\endgroup$
    – Douglas Zare
    Mar 30, 2011 at 20:47
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The way it works is not cumulative risk, but there is repeated occasions of risk over the period but no difference on each and every occasion. If the sky diver who goes once a year jumps on the same day your friend does, who's at the most risk? The answer barring variables, is they are equally at risk. However, your friend is exposing themselves to that risk multiple times, but that means only the possibility of the occurrence of accident prevents itself much more frequently. The comparison to a 6 rolled on a dice is not quite fair, it is not a certainty an accident occurs in skydiving, whereas, it is a certainty the dice has a 6, can roll a 6 and does so. So if you roll that dice there are going to be more 6s than there will be skydiving accidents in a 1000 attempts. If your friends risk increased with every attempt, then that means that an accident is inevitable and therefore every time not having one is making the next jump one closer to the accident, that or that accident occurrence is a changeable variable. The truth is it is a situation that unlike the dice, has many variables, the dice gets rolled, and even if there is a slope, a wind, water, a hard or soft throw, the dice will stop and it has only 6 possibilities. The skydive can have a rapid descent, slow, on target, off, have unknown buildings or objects in the path, birds, overhead cabling etc.

A skydiver who puts a parachute in a gym bag and jumps has increased risk of accident compared to the one who checks the chute, bag, altimeter, takes one or more reserves, has selected a landing site etc

So to answer the question. In this case, repetition does not increase his odds of having an accident unless he grows complacent

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