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I've ran in to a problem when trying to calculate a certain probability for a slot machine.

The slot maching works as follows: It has five reels (columns) and three rows. Each reels consists of 100 different "tiles" and 7 different numbers. The numbers are 0,1,2,3,4,5 and 20. The numbers DO NOT occur with the same frequency so to get 5 1's is not the same as getting 5 0's (probability wise).

The question is: if the slot machine shows 3 or more of the number 20 you will get a bonus game. What is the probabilty of this happening?

I tried making a sort of binary table where a 20 represents a 20 showing up and an X means it didnt.The included picture shows what I mean.Event table

I am not certain this is the right way to go though. I might just be completely wrong.

Any help would be greatly appreciated!

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    If I have understood this (not clear) then you see $15$ numbers on any trial, yes? Do you know how many of the tiles in a given reel show $20$? Do you know how they are distributed? – 2017-02-14
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    Looking at your event table it seems you only see $5$ numbers at a time. What, then, is the meaning of the "three rows"? – 2017-02-14
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    Yes it is 15 number on any trial and that's part of the reason I think my event table is incorrect. It was a long time ago I studied probability so I have a hard time determining just how to approach this. – 2017-02-14
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    Right. Well, the first question is "does the distribution of the $20's$ matter"? If, say, say there were two next to each other. Then, unless I botched it, there are $6$ possible spins that show you at least one $20$. But if the two $20's$ were opposite each other on the reel then there are $10$. Of course in the first case there are spins that show you more than one $20$ at a time. So...it seems you actually need to know the distribution on the reel. – 2017-02-14
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    As a suggestion: for real world problems like this, it is often easier to simply simulate the process to get the probability numerically. Nothing at all wrong with that. I expect it is possible to solve this particular problem exactly, given all the information. All you need is to know the probability distribution attached to each reel. That is the probability of spinning no $20's$, exactly one $20$, exactly two, exactly three. But the simulation might be faster and easier. – 2017-02-14
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    I will list the number of times a 20 occur on each reel. Reel 0: 5 times, reel 1: 3 times, reel 2: 2 times, reel 3: 3 times and reel 4: 5 times. I tried using these probabilities to make a probability table of each event from the event table but since the event table only looks at one line (5 numbers) and not three lines I assume it's wrong. I feel like I have solved it if the slot machine would only display 5 numbers at a time but since it's actually 15 im stuck. – 2017-02-14
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    As I say, that information is insufficient. You also need to know the distribution of the $20's$ on each reel. – 2017-02-14
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    Sorry, I thought pressing enter would put me on a new line and not add the comment I was currently writing. I am bit new to stackexchange. – 2017-02-14
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    It would be a lot easier if you only saw one number from each reel. As I say, though, if you insist on doing this analytically you will need more information. Specifically, for each reel you need to know the probability of showing twenty $0,1,2,3$ times. – 2017-02-14
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    I think I have to do it analytically since I don't have the programming skills to simulate it. But I will sit down with it some more and try to get something useful out of it. – 2017-02-14
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    If you can answer the question I gave, then the calculation is tedious but not terrible. It is easier, by far, to get the probability that you show no $20's$ out of the fifteen...then get the probability that you show exactly one, then the probability that you show exactly two. Good luck! – 2017-02-14
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    Good advice to calculate the opposite event! – 2017-02-15
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    Thank you for that advice! I had done the opposite before, calculating all the ways the slot machine could show 3,4 or 5 $20's$. I now tried to calculate the opposite event, take 1-P(opposite event), and it actually gave me the exact same result :) – 2017-02-16

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Probability table

Each row in the probability table corresponds to a row in the event table. So in the first row I have written down the probabilities that are associated with the first row in the event table.