# Thread: Calculating probability of the expected number of runs needed

1. ## Calculating probability of the expected number of runs needed

Hello, I'm working on a personal project but I've hit a bit of a wall in working out an equation. I'd like to find the probability of a result given certain constraints so I can estimate the expected number of runs needed to get a certain number of a result. In this case, I have a chain of probabilities where I have the probability of a result I'm hoping for. For each part of the chain, there is a probability of it returning something at all, then a probability of it being the result I want.

The first three will always return a result, 100% chance of a result, but there's only a 5% chance that this is the result I want. After these first three, I have five additional results. Each of these five have a 69% chance of occurring at all. They also will only occur if the previous result occurred. Along with that, they also only have a 5% chance of being the result I want. Now I have the equation to find the probability of getting my result that I want at least once, but I can't figure out the probability that it will occur any number of times, such as twice, or three times.

Hopefully I can explain this a bit clearer.

The first, second, and third result will always occur with a 5% chance of being the result I want.
The fourth result has a 69% chance of occurring, with a 5% chance of being the result I want.
The fifth, sixth, seventh, and eighth result will only occur if the the result before it has as well, with a 69% chance of occurring, with a 5% chance of being the result I want.

I don't often work with probabilities, so I hope I explained this clearly. Thanks in advance for any help, and please let me know if I should clarify myself on any part of this.

2. ## Re: Calculating probability of the expected number of runs needed

Let

be the number of successes in the first 3 trials.

be the number of successes on the possible next 5 trials.

With the assumptions of independence of the trials, we have follows a binomial distribution and follows a truncated geometric distribution.

And you would like to find the distribution of , which is the number of total successes.

The pmf of this is not very nice and I guess you need to calculate for each support point.

3. ## The Following User Says Thank You to BGM For This Useful Post:

Zackley (09-28-2015)

4. ## Re: Calculating probability of the expected number of runs needed

I was hoping that wasn't the case. I could calculate each probability manually, but in my personal project the probability of the result being the one I want is variable. There is always a 69% chance of the last 5 trials of occurring dependent on the previous success, but it's not always a 5% chance for it to be the result I want. I don't necessarily need the equation for those, as I'm working with a specific set of data, but it would have sped up the process.

Anyway, thank you for the help.

5. ## Re: Calculating probability of the expected number of runs needed

In the possible additional five rounds does it continue to go if anything occurred or only if the desired event occurs.

6. ## Re: Calculating probability of the expected number of runs needed

So long as any result occurs the rounds continue.

7. ## Re: Calculating probability of the expected number of runs needed

In those additional 5 rounds you say there is a 5% chance of getting what you want. Is that 5% of the time when something returns you'll get what you want or is it 5% chance to get what you want total?

Either way the approach will be like BGM said - we'd need to calculate for each support point. That actually isn't too bad though since we can get the pmf of X and Y fairly easily and getting the pmf of the sum is tedious to do by hand but isn't too bad to have a computer do it for you. I do these kinds of things in R a lot and that's not too bad but for others that don't like code I've shown how to do it in excel.

8. ## Re: Calculating probability of the expected number of runs needed

It's a 5% chance for each round. I'll code a function to figure out the probability of different amounts. Thank you for the help working this out though.

 Tweet

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts