 # Expected time for an event

Hi Experts,

An event can happen anytime during the next 10 years (2017-2026)
For each of the years 2017,2018,2019 we are given conditional probabilities of the event taking place. How can I solve for the probability of the event happening during 2020-2026 when my expected time of event is 2 years.

Thanks
Math / Science* probability Last Comment
d-glitch

8/22/2022 - Mon
d-glitch

What are the conditional probabilities you have been given for the first four years?
Can the event happen multiple times in the next ten years?
Must the event happen at least once in the next ten years?
Do you have any reason to believe this is a Poisson Process?
camper12

2017 - 15%
2018 (conditional probability) - 30%
2019(conditional probability) - 50%

The event can happen only once within the 10 years. This is why the conditional probability of the event happening in 2019 is conditional on the event not happening in 2017 and 2018.
The event actually may not necessarily have to happen within the 10 years.

I do not know how to think through whether it is a poisson process. Can you please help me with this?

Thanks
d-glitch

This is not a Poisson Process.  If it were, it could happen more than one time in 10 years.

What does this mean?
when my expected time of event is 2 years

In any case, with the info you have, you can calculate the probability that the event happens in  2017 [0.15]  or  2018 [0.85*0.30]  or  2019 [0.85*0.70*0.50].  Add those up and subtract from 1.000.
camper12

What it means that the expected time of occurrence is in 2 years and we need to determine the probability in 2020-2026 such that the expected time for the event to happen is 2 years.
d-glitch

I'm afraid that does not make any sense.
If this is a problem from a book, please post the exact wording.
If it is a problem from life or business or gaming, more detail might help.

Right now:  The probability for the event happening 2017 or 2018 is  0.405

If it does not happen this year:
As of 12:01AM on 01-Jan-2018:  the prob of the event happening in 2018 or 2019 will be 0.650
camper12

Hi,

Please find attached an excel. I am trying to find the expected time of the event. I have used solver so that expected time is 2 years. This is a real world problem.
Probability.xlsx
d-glitch

Well, it looks like you've solved it.  Are you happy with your solution?
I can't tell if it's any good because I don't know what you are trying to do or model or build.

But now Easter is approaching, and I may not be back until Tuesday.
camper12

I am not sure whether I solved it correctly. The objective is to get an expected value of 2 years!
d-glitch

If you set the probability of the event happening this year and every year to 0.5, then the expected value for the time until the event happens is always 2.0 years.

You can mess with the probabilities in the first three years if you really want, but it won't be as elegant.

Note that I solved for the Weighted Sum in Line 9 numerically using the Excel Solver and I truncated the infinite series after 20 terms.

``````But this can be solved analytically as well.

The Event Probability in Line 7 is a simple geometric series:
1/2  +  1/4  +  1/8  +  1/16  + . . .   =  1

The Weighted Sum in Line 9 can be expanded into an infinite sum of infinite
sums of geometric series:
1/2  +  1/4  +  1/8  +  1/16  + . . .   =  1
1/4  +  1/8  +  1/16  + . . .   =  1/2
1/8  +  1/16  + . . .   =  1/4
:                  :
:                  :

Note that this gives us one copy of the first term, two copies of the second,
and so on, as required.

Also note that the vertical sum is an another geometric series.
``````
camper12

Hey,

What if at the end of year 1 the probability of success is 15%; at the end of year 2 the conditional probability of success conditional upon the event not happening in year 1 is 30%. Similarly the conditional probability at the end of year 3 is 50%. Can you please help me understand how can I convert the conditional probability in years 2 and years 3 to apply this logic?

Also, can you please elaborate upon the following. I dont really understand.
The Weighted Sum in Line 9 can be expanded into an infinite sum of infinite
sums of geometric series:
1/2  +  1/4  +  1/8  +  1/16  + . . .   =  1
1/4  +  1/8  +  1/16  + . . .   =  1/2
1/8  +  1/16  + . . .   =  1/4
:                  :
:                  :

Thanks!
d-glitch

I am having some trouble with the way the problem is defined.  Are you talking about an event or a success?

After you have an answer to this question, what are you going to do with it?

This is from one of your earlier posts:
2017 - 15%     2018 (conditional probability) - 30%    2019(conditional probability) - 50%

This is from the most recent:
What if at the end of year 1 the probability of success is 15%
At the end of the year, won't we know if it happened?

I notice that the probability goes for the first three years?  Why.
What do you expect the probability to do for years four thru ten?  Does it continue increasing or level off?  Why.

What is the nature of this event or success?
How do know the the probabilities for some years but not for others?
Does the probability for the event happening this year take into account that a good part of the year has already passed?

What is your definition of expected value?  Is 2017  Year=1  or  Year=0  ???

At what point in time is the Expected Value for this process supposed to be 2?
camper12