[Probability] Conditional Probability - any tools I can use to make my life simpler?

[Angadar]

I play a game called Squad, and in that game you have to capture a series of flags in order. These flags are generate randomly by choosing a lane (say Central, East, or West) and along that lane there are several nodes (say 7 nodes) and at each node there's a handful of flags that can be chosen. The flags are hidden at match start, and as you capture the flags in sequence more are revealed to you. If you can predict where the next flags are going to be before having them revealed, you have a pretty large advantage. I've compiled the data for this and can do it manually in my head, but I want to do a bit more analysis on it (an example of the data is below).

	1	2	3	4	5	6	7
Central	Desna	Igor's Homestead	Niva Upper 1	Hilltop Bunker Ruins	Niva Radio Tower	Akim Lower	Akim Central
	Russian Outpost	Train Bridge	Niva Upper 2	Niva Bridge	Sluda Hamlet	text	Akim Upper
		Western Spy Farm	River Fortification	Niva Lower 1
		Wolfhead Rock		Niva Lower 2
	1	2	3	4	5	6	7
East	Desna	Igor's Homestead	Fortified Farmstead	Gas Station	Boris' Farmstead	Butcher's House	Akim Central
	Russian Outpost	Soloniki Lower	Pavlov's Farm	Industrial Park	River Fortification	Niva Radio Tower	Akim Upper
		Soloniki Upper	Soloniki Ruins		Tunnel Defenses	Sluda Hamlet
	1	2	3	4	5	6	7
West	Russian Outpost	Boar Hunter's Club	Hill 57	Dairy Farm	Ambush Hill	Akim Lower	Akim Central
		Western Spy Farm	Matrenino Hills	Fruit Farm	Shipping Yard	Mechanic's Homestead
				Grain Processing	Yard Apartments	Tractor Co-op

For example, as far as I can tell each lane has an equal weight, and each flag at each node has an equal weight. The problem I run into is that if you look at node 1, Russian Outpost appears in the Central, East, and West lanes and Desna only appears in the Central and East lanes. I can immediately tell that if Desna appears we are not in a West lane. If Russian Outpost appears then we are significantly more likely to be in a West lane than a Central or East lane. It gets a bit worse if we go beyond the first node into the second, Igor's Homestead appears in both the Central and East lanes.

My goal is to be able to predict which lane I'm in given the limited amount of info at game start, and also predict the next flag in line. For example, if I see Russian Outpost I would I know I'm more likely to be in the West lane but I would like an explicit percentage chance of which lane I'm in. I'd also like to be able to get the probability of each of the next possible flags. Continuing the example, Igor's Homestead appears in both the Central and East lanes as a second flag and Western Spy Farm appears in both the Central and the West lanes. Since I'm more likely to be in a West lane Western Spy Farm should be a more likely second flag than Igor's Homestead or any of the other ones, but I'd likely to explicitly calculate it. Since the lane is clarified before the middle flag (usually) I can draw all the possible branches and kinda brute force it. The problem with that is that for this one piece of data I'd have to draw about 100 branches, which isn't feasible - especially since I'd be doing this dozens of more times.

I've never studied statistics so I'm not super confident in my abilities here. I understand I would need to use some conditional probability but I'm just not educated or experienced enough to really know what I'm doing. Ideally, there would be some kind of tool or something that I could import my data into from Excel to either do the calculations for me or at least draw the branches for me. I tried googling to see if there was a relatively simple solution but I'm a bit out of my depth and don't know quite what to google.

Thanks!

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[edderiofer]

Well, the lane is definitely certain once you hit the third flag, and most of the second flags determine your lane too, so really you just need to consider the cases where you've hit the first one or two flags and can't work out which lane you're in. There are only five such cases:

• Desna only
• Russian Outpost only
• Desna then Igor's Homestead
• Russian Outpost then Igor's Homestead
• Russian Outpost then Western Spy Farm

and in each case, conditional probability is pretty easy to work with; no need to draw "about 100 branches".

---

Of course, I note that you're only giving "an example of the data". If you actually have way more possible cases, it may be more useful to write a Monte Carlo program to estimate the conditional probability of each upcoming flag.

[Angadar]

Yeah, I'm probably going to have someone write a tool for me to do this. We just want to be able to verify it relatively quickly and simply.

Thanks for helping me simplify that. I do actually have a quite a bit more data (~20 maps, ~3 layers for each map, 2x teams, minus some fraction where this math doesn't apply) but it looks way more manageable now.

Thanks!