What are the chances ?

Ben has been monitoring earthquakes in Oklahoma for about 10 days now ... in that time there have been about 8 earthquakes > M3   and about half of these have been registered on our trusty geophone.

(seismologists look away ! our window-sill sensor on the third floor !)

  But how long will Ben have to wait until he feels an earthquake ?   Chatting to people in the hotel it seems that  M5+ events even 160+ km away cause shaking strong enough to be easily felt.

It turns out that there is a pretty good rule of thumb that seismologists can use to estimate how frequently earthquakes occur in a given location (seismologists call it the Gutenberg-Richter Law ) It seems that for most places for most of the time earthquake size and frequency of  occurence follow a simple power of 10 rule (roughly).   For every decrease in magnitude of one unit, the number of earthquakes increases by a factor of 10.  So if there have been 8 earthquakes >M3 in the last 10 days there were probably 80 earthquakes >M2  (unfortunately we cannot detect earthquakes this small with our geophone and even the Oklahoma Geological Survey only guarantee measuring earthquakes  >M2.5 )

Looking back over the past year there have been 567 earthquakes >M3 in Oklahoma  37 >M4 and 3 >M5

The past year's earthquakes >M3 in Oklahoma 

(You can have a play at making your own maps at http://ds.iris.edu/ieb/ )

so you would expect there to be an earthquake >M5 every 100 days or so,

But Ben is only planning on being in Oklahoma for 70 days so does this mean he won't feel one ?   Well not necessarily ... earthquakes tend to occur at fairly random times so we need to start thinking about probabilities ...  Time to do some MATHS !!! 

Let's start by thinking about the probability of a M5+ earthquake occuring somehwere in Oklahoma state today as 1 in 100 (or 0.01) .... this means that the probability of an earthquake not occurring to day must be 99 in 100 (or 0.99) .  (It turns out that measuring the probability of an earthquake not occurring is a more useful way at solving this problem) 

Now if we consider the probability that two consecutive days won't have a M5+ earthquake we have to multiply the probability of this happening on each day so the probability is 0.99 x 0.99 = 0.9801 

So the probability that an earthquake not happeneing for 70 consecutive days should be 

0.99 x 0.99 x 0.99 .........   (70 times)   which is 0.99 ^70   = 0.494   i.e. roughly 50%   or 1 in 2 

So if there is a    50 % probability that an earthquake >M5 will not happen for 70 consecutive days then there will also be a 50% probability that there will be an earthquake >M5 occuring in a consecutive 70 day period.