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- TL;DR Summary
- I have a task, the conditions of which you can read below. I understand how to solve it by approximating the value, but I would like to know how I can derive an equation (and is it even possible?), which can calculate the exact value.

Hello! A friend shared a problem he recently solved. It goes as follows:

How to solve this problem through approximation is obvious to me. Here's the Python code:

The answer is approximately 31.5.

However, I am interested in the possibility of solving this problem analytically. That is, to write some equation that can be calculated for n values and get not an approximation, but an exact answer. Is this possible? I am weak in mathematical statistics and cannot imagine how to do this. You can simply point out some vectors of study that I need to explore to derive the equation myself, but I would be more grateful if you could derive this equation.

Each dagger strike deals either normal damage = 20 or critical damage = 80.

After each strike, the probability of dealing normal and critical damage changes (initial probabilities are 90% and 10% respectively). The probability changes according to the following two conditions:

1. If the previous strike dealt normal damage, the probability of dealing normal damage decreases by 3%, and the probability of dealing critical damage increases by 3%.

2. If the previous strike dealt critical damage, the probability of dealing normal damage becomes 90%, and the probability of dealing critical damage becomes 10%.

The average damage for n strikes.**Given:**Each dagger strike deals either normal damage = 20 or critical damage = 80.

After each strike, the probability of dealing normal and critical damage changes (initial probabilities are 90% and 10% respectively). The probability changes according to the following two conditions:

1. If the previous strike dealt normal damage, the probability of dealing normal damage decreases by 3%, and the probability of dealing critical damage increases by 3%.

2. If the previous strike dealt critical damage, the probability of dealing normal damage becomes 90%, and the probability of dealing critical damage becomes 10%.

**Find:**The average damage for n strikes.

How to solve this problem through approximation is obvious to me. Here's the Python code:

The answer is approximately 31.5.

However, I am interested in the possibility of solving this problem analytically. That is, to write some equation that can be calculated for n values and get not an approximation, but an exact answer. Is this possible? I am weak in mathematical statistics and cannot imagine how to do this. You can simply point out some vectors of study that I need to explore to derive the equation myself, but I would be more grateful if you could derive this equation.