Slot Machine Probability
Hello. My problem involves a slot machine that has 5 reels and 4 rows. Each reel is 131 symbols in length and the symbol count of each reel is in the Excel doc in the Dropbox link below. The questions involve the frequency of certain symbols landing within the 5x4 grid. There are 8 total questions, however, I believe most can be solved using the same processes.
https://www.dropbox.com/s/4tst1rpq16lzqtu/Example%201.xlsx?dl=0
Only 1 "Free Spin" symbol can land in a given reel at a time and the row the "Free Spin" symbol lands on within the reel does not matter.
 What is the probability of getting 3 "Free Spin" symbols within the Main Game?
 How many spins on average does it take to land 3 "Free Spin" Symbols within the Main Game?
It is possible for 0, 1, 2, 3, or 4 "Block" symbols to land in a given reel at a time. So the least amount of "Block"s that can land across the entire slot machine is 0 and the most is 20. For example you can have 1 "Block" in Reel 1, 2 "Block"s in Reel 2, and 1 "Block" in Reel 4 and that will count as 4 "Block"s across all of the reels. Order and position do not matter.
 What is the probability that a total of 4+ "Block"s land across all of the reels within the Main Game?
 How many spins on average does it take to land 4+ "Block" symbols within the Main Game?
 What is the probability that a total of 4+ "Block"s land across all of the reels within the Free Spins Game?
 How many spins on average does it take to land 4+ "Block" symbols within the Free Spins Game?
 What is the probability that a total of 4+ "Block"s land across all of the reels for both the Main Game and Free Spins Game?
 How many spins on average does it take to land 4+ "Block" symbols across both the Main Game Free Spins Game?

I have some clarifying questions: Are the main game reel and the free spins reel spun at the same time? Do they spin independently of one another? Are the rows independent? Like if reel 1 row 1 is on position 0, does that mean that row 2 HAS to be position 1, row 3 has to be position 2, row 4 has to be position 3?

For this question: "What is the probability that a total of 4+ "Block"s land across all of the reels for both the Main Game and Free Spins Game?" does the main game need a free spin in order for this to happen?

Another question: Do Wilds count as Free Spins? Do Wilds count as Blocks?

Hello. Thank you for your questions and sorry for my ambiguity. The main game reel and free spins are not spun at the same time. The main game is played until 3 free spin symbols occur and then the free spin game is played for 10 spins then it returns back to the main game. Reels 1 through 5 spin independently. And yes if reel 1 row 1 is on position 0, row 2 has to be position 1, row 3 has to be position 2, row 4 has to be position 3.

The main game is played until 3 free spin symbols land. When 3 free symbols land, the free game reel strip is played for 10 spins. Since the main game and free spin reel strips have a different amount of blocks, my question is about the combined probability of the two since I assume the probability of getting 4+ blocks increases from the probability of just the main game.

Wilds do not count as Free Spins or Blocks. Thank you for that question. It shows that you know how games of chance generally work with regards to wilds :)

Also, just wanted to reiterate one thing since my terminology isn't the strongest. For example, reel 1 row 1 can be randomly on position 52, and reel 2 row 1 can be randomly on position 100, etc. However, in this scenario, reel 1 row 2 must be on position 53, and reel 2 row 2 must be on position 101.

When you say "until 3 Free Spins land" do you mean on a single pull of the lever? So you can't get 2 on the first spin and then 1 on the second spin, and then you have accumulated 3, so now the Free Spin game starts? All 3 have to occur on the same spin? For all of your questions, are you interested in getting a particular number of symbols on a single spin, or accumulate those symbols over the course of several spins?

also, I'm still trying to understand question 5. "a total of 4+ "Block"s land across all of the reels for both the Main Game and Free Spins Game" But the Free Spin game taking place at all depends on what happens in the main game, right? So you would need 4+ "Blocks" in the main game AND 3+ "Free Spins" so that the Free Spin game happens, AND THEN you need 4+ "Blocks" in the Free Spin game? Sorry, this question is particularly confusing. The rest should be pretty straight forward to compute.

I am interested in getting a particular number of symbols on a single pull of the lever for all of the questions, not over the course of several spins. So that is correct you can't get 2 "Free Spins" on the first spin and then 1 on the second spin. Same with the "Blocks." Having a total of 4+ across the board on one spin is what counts. Having 2 blocks land on one spin then 2 blocks land on the next spin does not count.

For your second question, here is an example with made up numbers and I hope I'm able to clarify. Let's say the probability of landing 4 blocks on one pull of the lever in the Main Game is 1/100. Then let's say you get a free game on average every 200 pulls of the lever which triggers the Free Game reel strip for 10 spins. Since the Free Game reel strip has more blocks per reel, the probability of getting 4+ would be greater than the Main game for those 10 spins.

Let's say the probability of getting a block in the Free Game reel strip is 1/50 for those 10 spins. I guess would the probability of getting 4+ blocks be the weighted average of the probability of 4+ blocks in the main game added to the frequency of 4+ blocks in the Free Spins game?

Please let me know if you need any more clarification or a deadline extension. I'm happy to accommodate.

"I guess would the probability of getting 4+ blocks be the weighted average of the probability of 4+ blocks in the main game added to the frequency of 4+ blocks in the Free Spins game?" I don't know why you want to add them. If anything, you would multiply them. If a Main game and a Free game are independent, then P(4+ Block on Main game AND 4+ Block on Free game) = P(4+ Block on Main game)*P(4+ Block on Free game)

What I still don't get is that getting a Free game at all DEPENDS on the Main game. So it's still not entirely clear to me the exact event you're asking the probability of.

Hmm, sorry for my lack of clarity. It is possible that question 5 and 6 don't entirely make sense mathematically when I thought they did when writing them out. Please disregard questions 5 and 6. I will be happy to accept an answer fully without an answer for those two.
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Thank you for going above and beyond with your answer! This all makes sense to me except for sheet 1 of your Excel doc. I see there are 131 positions with 5 reels which makes sense, but I am confused by the grid within. What do those numbers represent? Also, one of the questions you asked me is if reel 1 row 1 is on position 0, row 2 has to be position 1, etc. Is that represented in the numbers (or is that even pertinent to the solution if pulls are random)?

The numbers on sheet 1 represent “how many of a particular symbol are visible when a particular reel is in a particular position.” Like if reel 1 is in position 0, you can see whatever symbols 0 through 3 are. If it’s in position 23, you can see whatever symbols 23 through 26 are. Among those symbols there are either 0, 1, 2, 3, or 4 of the particular symbol you care about (either Free or Block). That’s what the numbers on sheet 1 are.
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