How likely is that?
A random event occurs as an average once per year. How likely is it that it occurs a given year?
Problem source: Rodolfo Valeiras.
A random event occurs as an average once per year. How likely is it that it occurs a given year?
Problem source: Rodolfo Valeiras.
April 9th, 2009 at 3:09 am
There is not enough information. For example, perhaps it occurs in any year with probability 1%, and when it does occur, it occurs 100 times that year.
April 9th, 2009 at 6:23 pm
I interpret “on average once a year” as “each day there is a 1/365 chance that the problem occurs”. You can take it from there.
April 9th, 2009 at 8:08 pm
The usual model for counting processes is the Poisson model–used for things as diverse as radioactive decay and fatalities from horse kicks. The model is
Pr(#events = k) = exp(-rate) rate^k / k! where k = 0, 1, 2, …
With a rate of 1/year, the probability of exactly one in a year is exp(-1), and the probability of AT LEAST one in a year is 1-exp(-1), since the probability of NO events and of ONE event are equal when the rate is 1.
Of course, this is only a model, albeit one that’s been used successfully for over a century.
April 12th, 2009 at 6:17 am
Depends on what you mean by “random” here. There’s an infinitude of ways for a random event to average once per year. You must be precise for the question to be meaningful.
One such nterpretation leads to the Poisson, so the probability of exactly one event would be 1/e, and of at least one event 1-1/e.
April 14th, 2009 at 6:19 pm
Just for fun:
My naive approach detailed above gives as answer
1 – (1-1/365)^365
1) My naive interpretation gives a probability of about 0.632625
2) The Poisson interpretation gives a probability of about 0.632121
Is it a coincidence that they are so close? No! I “discretized” the Poisson process, by considering an event as an entity that operated by days. If I had discretized by hours, or minutes, or seconds, the two answers would have been closer still!
I am, in fact, approximating an integral by the “midpoint rule”.
October 18th, 2009 at 11:03 pm
I agree in principal with Richard Sabey’s statement. But If the event happening once led to it happening another 99 times, I don’t feel comfortable with 99 of those events as being independently random. Not that the word independently was used in the question. If the 100 events came together, I’d regard the event as being the collection of 100 sub-events. So 1/e for me too.