Percentage/chance to make certain amounts of gross income from a game

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Nubbly

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This is not a homework or school related problem, but I wanted some help with it. It revolves around a money making method I have in a game that I play, and what chances I have to make certain amounts of money. I just never learned, understood or went into probability and statistics, so that's why I came here to ask.

So the information I have is here:

I can purchase a "gift tree" seed for 1,900,000 gold (1.9M) and plant it. It has a chance to give me either 2, 3, 4 or 5 "gift fruit" in return. Since there is 4 outcomes, each has a 25% chance of occurring. A single "gift fruit" can be sold for 700,000 gold. (700k)

I know that I have a 75% chance to make profit, as getting 3, 4, or 5 fruit give me back more money than I spent, receiving 2.1M, 2.8M or 3.5M respectively. I also have 20 planting spots to plant trees, so this happens 20 times.

What I want to know is what percentage/chance I have to lose money, and the percentage/chance I have to make a total of 40M, 50M, 60M and 70M respectively in gross income from 20 planted trees. I already know that by buying 20 trees, I have a very high chance to make profit, I just don't know what that chance is.

I honestly just want to understand how to calculate that, as I have no idea and I find it very interesting. I hope I gave enough information, and I would appreciate an explanation or help, thanks for reading :)
 
Nubbly said:
This is not a homework or school related problem, but I wanted some help with it. It revolves around a money making method I have in a game that I play, and what chances I have to make certain amounts of money. I just never learned, understood or went into probability and statistics, so that's why I came here to ask.

So the information I have is here:

I can purchase a "gift tree" seed for 1,900,000 gold (1.9M) and plant it. It has a chance to give me either 2, 3, 4 or 5 "gift fruit" in return. Since there is 4 outcomes, each has a 25% chance of occurring. A single "gift fruit" can be sold for 700,000 gold. (700k)

I know that I have a 75% chance to make profit, as getting 3, 4, or 5 fruit give me back more money than I spent, receiving 2.1M, 2.8M or 3.5M respectively. I also have 20 planting spots to plant trees, so this happens 20 times.

What I want to know is what percentage/chance I have to lose money, and the percentage/chance I have to make a total of 40M, 50M, 60M and 70M respectively in gross income from 20 planted trees. I already know that by buying 20 trees, I have a very high chance to make profit, I just don't know what that chance is.

I honestly just want to understand how to calculate that, as I have no idea and I find it very interesting. I hope I gave enough information, and I would appreciate an explanation or help, thanks for reading :)
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This isn't a homework problem or even school related, I haven't "tried" anything. It's just something I thought of and don't know what to do with. I don't know how to go about solving it at all, so I came here to ask. This "problem" is purely recreational.
 
All right. I take you at your word so I shall give you extra help, but the only way to understand a concept is to do some work yourself.

First, you are making a very dangerous assumption. You are assuming that because there are four possibilities, each has a probability of 25%. Different possibilities may have different probabilities. Do you think that the probability of the casino making money is the same as the probability of the casino losing money?

In the case of this kind of game, however, the assumption is plausible so let's work with it. Solving some of the questions you are asking calls for a lot of arithmetic. You can get a computer program to do the arithmetic if you know how to do that. For example, do you know how to use excel?

Let's start with an idea that does not involve a lot of arithmetic. It is called expected value. It tells you what your net benefit will likely approximate if you play many times.

[MATH]0.25(1.4 - 1.9) + 0.25(2.1 - 1.9) + 0.25(2.8 - 1.9) + 0.25(3.5 - 1.9) =\\ 0.25(-0.5 + 0.2 + 0.9 + 1.6) = 0.25(2.2) = 0.55.[/MATH]This tells you that if you plant a single tree many times you can expect to average about a net gain of 550 thousand over time. What is nice about expected values is that they add. So if you plant 20 trees many many times, you can expect to average a net gain of about 11 million over time.
 
JeffM said:
All right. I take you at your word so I shall give you extra help, but the only way to understand a concept is to do some work yourself.

First, you are making a very dangerous assumption. You are assuming that because there are four possibilities, each has a probability of 25%. Different possibilities may have different probabilities. Do you think that the probability of the casino making money is the same as the probability of the casino losing money?

In the case of this kind of game, however, the assumption is plausible so let's work with it. Solving some of the questions you are asking calls for a lot of arithmetic. You can get a computer program to do the arithmetic if you know how to do that. For example, do you know how to use excel?

Let's start with an idea that does not involve a lot of arithmetic. It is called expected value. It tells you what your net benefit will likely approximate if you play many times.

[MATH]0.25(1.4 - 1.9) + 0.25(2.1 - 1.9) + 0.25(2.8 - 1.9) + 0.25(3.5 - 1.9) =\\ 0.25(-0.5 + 0.2 + 0.9 + 1.6) = 0.25(2.2) = 0.55.[/MATH]This tells you that if you plant a single tree many times you can expect to average about a net gain of 550 thousand over time. What is nice about expected values is that they add. So if you plant 20 trees many many times, you can expect to average a net gain of about 11 million over time.
Click to expand...
The 25% chances were both confirmed by the developers, I didn't not assume that they would each be 25%.

I've heard of expected value, but was never taught or had to use it, but I understand how it works, and it checks out with the results I've gotten.

And no, I never actually had the opportunity to learn or use excel, but better late than never. q:
 
So, expected value is the sum of the products of each possible gain (positive) or loss (negative) and its probability. Your gain or loss over n trials if the expected value is e will very likely APPROXIMATE n times e if n is a large number.

To find out exact probabilities of specific outcomes with multiple trees is much more computationally burdensome. Let's think about two trees rather than twenty and assume, which is plausible in this case, that the probabilities for each tree are unaffected by the number of other trees (another frequently dangerous assumption). Now there are 16 possible outcomes (4 possibilities for one tree, and for each of those possibilities, 4 possibilities for the other tree). Each is equally likely. With 3 trees, the number of possibilities is 64. With 20 trees, the number of possibilities is 4^20, which is about equal to
1,000,000,000,000.

Fortunately, there are ways to reduce this computational nightmare. One is to see that which specific tree does what is irrelevant to the total gain or loss. What is relevant is how many trees do what. So in the case of two trees, the relevant possibilities are as follows

4 gifts, probability 1/16
5 gifts, probability 2/16 (2 on tree 1 and 3 on tree 2 or 3 on tree 1 and 2 on tree 2)
6 gifts, probability 3/16 (why?)
7 gifts, probability 4/16
8 gifts, 3/16
9 gifts, 2/16
10 gifts, 1/16

Did you follow that? We reduced the number of relevant possibilities from 16 to 7, but we no longer can assume equal probabilities. Instead, we need to figure out the probabilities for each of the seven cases, but they turned out to follow a sort of neat pattern

Give it a go for three trees. See if the same kind of pattern emerges. This is how you do math.
 
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