How do I compute the cumulative probabilites of multiple bell curves?

[k_squared]

I took statistics in university about two years ago, but I'm rusty. I was trying to write a zero player game - except sometimes, the player can control one of the characters, and I needed to be able to compute these probabilities. That said, I almost put this in homework help, but it is not homework. Haven't been to school in years (you can see when I last posted there.) It took me a while to come up with this, it's basically everything I think I need to know to run the simulation.
I'd be just as happy if you give me somewhere I can look this up than a direct answer.

Let's say we are going to a trip to some ancient ruins. 90% of trips will be safe, but 10% of trips will have someone be chased by an angry mummy. If the trips are all two weeks, I can do x14=9/10 to find the chances of NOT being chased by a mummy; on any given day then 1−x is the actual probability in a day that you will trip an ancient curse.

What I think I'm asking, though, is how to "superimpose" bell curves. I'm not sure if that is the proper wording for this operation, though.

Let's now say that:
1.) The average trip is two weeks.
2.) 90% of trips will end within two days of two weeks (can I use that to find σ?)

How do you compute the chances of the trip ending on any given day?

3a.) 40% of the trips end on a Saturday. On the other days of the week, the chances are 10% that the trip will end. OR:
3b.) The most common time of the trips to go home at Saturday at 1:00 p.m., with a standard deviation of 30 hours.

Now how do you compute the chances of the trip ending on any given day?

4.) We still know from the first sentence that in 10% of the trips someone will wind up being chased by a mummy.

What are the chances given the numbered conditions above, on any give day, that someone winds up being chased by a mummy?

I think these scenarios cover all the types of computations I need to do, but I'm not sure if we covered all of these in class or not.

 

[Dale]

Do you want a 16 day trip to have the same mummy probability as a 12 day trip? I would think that longer trips should have higher mummy probabilities.

Edit: if that is correct then you can simply use a fixed probability every day or hour or whatever such that 90% of the 14 day trips have no mummies (with $X$ a binomial random variable $X\sim B(n,p)$ solve $P(X=0|n=14,p)=0.90$ for p). If I did it right that is p=0.0075 chance of a mummy every day.

 

[Stephen Tashi]

I took statistics in university about two years ago, but I'm rusty.

I'd be just as happy if you give me somewhere I can look this up than a direct answer.

An introductory statistics course won't have covered all the questions you have asked. It may have covered questions of the form: "The distribution of number of days is ...<such and such> and probability of a mummy chase is <so-and-so> per day. What is the expected probability of a mummy chase?" That type of question involves "conditional expectation" It's a topic that you can look up.

To answer the complete variety of questions in your post,
What you need to do is the general task of formulating a "probability model". This amounts to creating a detailed simulation. The simulation could be a computer algorithm or a set of instructions about how the "game" can be simulated manually using charts and dice. If you fail to do this then your questions are ambiguous because they aren't specific mathematical questions.

Some comments on your post that emphasize that point:

Let's say we are going to a trip to some ancient ruins. 90% of trips will be safe, but 10% of trips will have someone be chased by an angry mummy. If the trips are all two weeks, I can do x14=9/10 to find the chances of NOT being chased by a mummy; on any given day then 1−x is the actual probability in a day that you will trip an ancient curse.

You are using the term "trip" in an inconsistent manner. If there is a 0.9 probability that a "trip" is safe, then you shouldn't use 0.9 as the probability that a "day" is safe.

Is there a chance the mummy catches the person and kills them? - thus making all questions about subsequent trips and days moot?

Let's now say that:
1.) The average trip is two weeks.
2.) 90% of trips will end within two days of two weeks (can I use that to find σ?)

In your notation, what is the definition of $\sigma$? Does this have to do with being chased by a mummy or does it have to do with the length of a trip?

How do you compute the chances of the trip ending on any given day?

3a.) 40% of the trips end on a Saturday. On the other days of the week, the chances are 10% that the trip will end. OR:

You haven't proposed any probability model (i.e. detailed method of simulating) how trips begin and end. You have only stated a desired property of such a model. Many different models could have this property. It isn't clear what model to use in computing answers to your questions.