If there's a 99% chance of earthquake in the next 30 yrs and a 99% chance my mail will be delivered by 6pm...

Question

All right. So let's say, like a recent news article does, that there's a 99% chance California will have a major earthquake in the next 30 years. Then say we're in the last week of year 29, and there's been no major earthquake yet. What's the chance of their being a major earthquake during that week?

Then say I'm having a package delivered and, because the mail is really regular where I live, there's a 99% chance the package will be here by 6 PM. When it's a few minutes shy of 6 PM, isn't it reasonable for me to assume that the package will probably show up in the next few minutes? But I wouldn't make an assumption like that about the earthquake coming within that last week.

What's the difference between these two cases? I'm try to understand how the statistics works here.

Answer 1

There's a lot involved in the probabilities your are asking about.  

The mail example may have a lot of conditions or dependencies on your probability.  For example, if I tell you I will do something by 6:00, and I am very reliable, then the closer it gets to 6 and I haven't done it, the better the chance of me doing it.  That's because I am changing my behavior as it gets later--as my watch is clicking towards 5:45, I'm thinking "I gotta get this package to her ASAP!" and I rush.  You know that my behaviour is dependent on how close it is to 6:00pm.  

Let's look at another example of independent events and probabilities.  If I flip a coin every hour for 12 hours, I could say there is a 99.98% chance that I will flip at least one heads.  that's the inverse of the probability of flipping no heads, or 1-(1/2)^12=.999756.  but in this case, each flip is independent of each other.  The behavior of the coin does not change based on the time of the day.  so at the last coin flip of the day, if I haven't flipped a head yet, my probability of flipping a head is 50% as always, not 99.975%.  

The big question is whether an earthquake is like a mailman or like a coin.  I'd say there's a little bit of both.  Over a short time scale, earthquakes are totally random--they're like a coin flipping.  But geologists know that over a long time scale, pressure is building up at faults between tectonic plates, and as the pressure builds, the stress eventually is greater than the yield stress of the rock and it breaks, causing an earthquake.  Over long time scales, the probability of an earthquake increases, kind of like the mailman.  The problem with your statistics question here is that we do not know enough information on how dependent or independent the probability of an earthquake is over time and at what time the event is more or less certain.  Even if it is certain over a span of time, it might follow a poisson distribution, which explains the probability of random things happening within a certain time. There is no magic number of exactly 30 years or anything, they're just looking for somewhere near the tail end of the distribution.  I'm not an expert at time distributions, but I hope that helps a little to show the dependence and independence of events.

Answer 2

this might be way off, but;

it's easier to figure out by figuring out if it WON'T happen:

30 years = 1560

probability per week = 0.99/1560

probability of it not happening per week = 1-(probability of it happening)

probability of a specific week = probability of it not happening raised to the power of the week number.

1-((1-(0.99/1560))^1559) = 0.6283041812538149853

62.83%  (I think). Obviously this is not realistic, if you asked someone a week before the thirty years was up, they wouldn't say "there's a more than 60% chance that we'll have an earth quake in the next 7 days". Note that using the same equation, at the end of thirty years, the probability is

0.62854006513878852579 = 62.85% of it happening that week. After 60 years, it's 0.86201751679290452852 = 86.2%.

Same for your mail, the closer it gets, the higher the probability of it coming before 6pm, but it'll never reach 100% (because after 6pm, it won't matter)

Answer 3

For the sake of this discussion, lets consider an earthquake and it's associated aftershocks as a single event.  This would not be unreasonable as most aftershocks are not themselves considered major.

An earthquake is the result of the release of accululated stress built up from 2 plates moving aginst each other.  The plates are moving at a fairly constant rate.  The line where the two plates meet is the fault line.  The probability of an earthquake happening is not independent.  This is because the release of the accululated stress at the fault line is what an earthquake is.  Once that stress is relieved an earthquake is far less likely.

http://www.earth.northwestern.edu/people...

I think here you are trying to learn about statistics and not so much about earthquakes and mailcarriers.

That said, let's change "earthquake in California" to "hurricane in Florida".  Also, let's use your same probabilities - 99% chance a hurricane will hit city "X" sometime in a 30 year period.  Annually this is independent of whether the city was hit last year or not.  The formula for the probability the city will be hit in t years is:

1-(1-.99)^(t/30)

years --- prob

30 ------ .99

15 ------ .90

10 ------ .78

5 ------ .54

1 ------ .14

Now let's look at mail delivery.  The mailcarrier's arrival time can be aproxmitated with a normal distribution, also known as a bell curve.  You had stated that there was a 99% chance the mail would arrive before 6:00.  Let's add to that by stating that there is also a 99% chance it will be delivered after 4:00.  With a mean arrival time of 5:00.  The times before 4:00 and after 6:00 is called the tails.  There is a 1% probability the mail will show up before 4:00, a 1% probability it will show up after 6:00 and a 98% chance it will be delivered between 4:00 and 6:00.

Let me know if that didn't answer your question.