Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

**Fruityloop****Member**- Registered: 2009-05-18
- Posts: 120

I was thinking of a math problem that I've never seen before.

Let's say you have rainfall (or temperature) records at a certain location going back 80 years.

On average, how many years will it have been since the last record has been set?

I tried doing the math for 5 years and I'm not certain if it's correct or not.

My reasoning is as follows...

The probability of the first year record still standing after 5 years is because each of the following 4 years has 50% chance of breaking it or not.

The probability of the second year setting a new record is and the probability of it still standing after the following 3 years is

And so on...

Is this correct or not?

The eclipses from Algol (an eclipsing binary star) come further apart in time when the Earth is moving away from Algol and closer together in time when the Earth is moving towards Algol, thereby proving that the speed of light is variable and that Einstein's Special Theory of Relativity is wrong.

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,663

Hi Fruityloop;

because each of the following 4 years has 50% chance of breaking it or not.

I would question that. Each record would be harder and harder to break. Also, setting the first probability to 50 % is arbitrary.

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

Offline

**Fruityloop****Member**- Registered: 2009-05-18
- Posts: 120

Thank you for your response Bobbym.

This is kind of a strange problem because we are dealing with something varying around an average that is unknown.

True, the setting of 50% is somewhat arbitrary, but the first year that there is a record we assume that the rainfall (or temperature)

record is just as likely to be above as below the average, since we don't know what the average is.

because each of the following 4 years has 50% chance of breaking it or not.

I'm comparing the following 4 years only to the first year.

There are 2 possible sequences for the first 2 years, with only one sequence having the 2nd year setting a new record. So we have a probability of 1/2 for the 2nd year setting a new record. Similar reasoning shows that the probability of the 3rd year setting a new record is 1/3 and so on...

*Last edited by Fruityloop (2013-09-11 14:59:37)*

The eclipses from Algol (an eclipsing binary star) come further apart in time when the Earth is moving away from Algol and closer together in time when the Earth is moving towards Algol, thereby proving that the speed of light is variable and that Einstein's Special Theory of Relativity is wrong.

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,663

Hi Fruityloop;

What are your possible sequences for 3 years?

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

Offline

**Fruityloop****Member**- Registered: 2009-05-18
- Posts: 120

I feel like such a ding-dong. I think I know the answer now.

Let's say that there are records going back N years. The probability of each year being the one that holds the record is 1/N.

So the average number of years since the latest record (which is the record overall of course) is

I generated 65 random numbers between 1 and 1300 inclusive and did this 50 times.

The average number of numbers since the all-time high number was 30.28 which is close to 33 which is what we would expect.

Let's say you have temperature records going back 130 years, the average number of years since the all-time high

record would be 65.5. To me this seems counter-intuitive because it seems like later years would be more likely to hold

the all-time high records.

*Last edited by Fruityloop (2013-09-18 09:44:22)*

The eclipses from Algol (an eclipsing binary star) come further apart in time when the Earth is moving away from Algol and closer together in time when the Earth is moving towards Algol, thereby proving that the speed of light is variable and that Einstein's Special Theory of Relativity is wrong.

Offline

**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,658

How is that probability 1/N? It seems rather arbitrary.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

Offline

**Fruityloop****Member**- Registered: 2009-05-18
- Posts: 120

Each year has an equal probability of being the record-holder. There are N years. So for each year the probability of it being the record-holder is 1/N. Let's say you have N balls in a bag, they are numbered 1 to N. You reach into the bag and pull out a ball at random and without looking at the number on the ball, you write a random number between 1 and 1500 on the opposite side of the ball. You do this for all of the N balls. Which numbered ball will have the highest number? It is easy to see that each ball has an equal chance of having the highest number.

Offline

**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,658

What if none of them is a record holder? And, besides, it's something you can only guess to have a probability you think it has.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

Offline

**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,532

Fruityloop wrote:

Each year has an equal probability of being the record-holder.

I don't agree with that assumption. With climate changes, for example, if, say, it is getting wetter, then the probability of a new record gets higher as each year passes, and diminishes if it is getting dryer.

The year that records began had lots of broken records. What a bumper year that was!

Even a low record still counted as the new record. So it was easy to break it. As the record values increase, it becomes harder and harder to break them even if there is no trend in climate change.

If, let's say, the rainfall in any year is part of a normal distribution, then you could estimate the chance of getting a high value one particular year. Let's say that is the new record. Then for subsequent years, you have to get new value that is beyond that value in order to set a new record. The 'tail' of the distribution will diminish over time.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Offline

**Fruityloop****Member**- Registered: 2009-05-18
- Posts: 120

Thanks for the responses. I was assuming an unchanging climate in my problem.

I agree that with the ever-increasing temperatures, it is becoming more and more likely to set new higher temperature records.

My idea was to figure out what would be an average number of years since a new temperature record was set and then

take an average for all 365 days from a weather station and do a comparison. When I have time maybe I will do that.

Offline