Board Thread:General Discussion/@comment-26289928-20151115091932

I truly do not have a life ᕕ(ᐛ)ᕗ If I feel like it I might do the same for the rest of the event maps w

So some JP player calculated the probability of drawing an enemy card on each node for map e4 of the Village of Treasures Practice event (see here) so I decided to give it a shot for the current event.

Please note that this % calculated is theoretical, meaning that if you wanted a % more relevant to you, you have to collect data of your own enemy card draws. That being said, if someone has kept a table/spreadsheet of said enemy card draws, please do send them my way/reply to this thread. I want to see how accurate this equation is.

I know most of you don't want to live/scroll through the entire explanation, so here you go, the rate of appearance: 69.3911464%

For those who are curious as to the math behind it, I'll try my best to explain here. I based my calculations off the JP user's equation in the aforementioned post. Here is the equation I got:

https://i.gyazo.com/e320dd626ec6eb9c4ed51f47f977180e.png

Wtf? What the flying cowpat is this? It's actually not that complex, in essence this is just basic probability written in a simplified way to prevent it from becoming the length of the Great Wall of China.

The probability of something is https://i.gyazo.com/5f707fa6395690e4ce13e0a9ef441b32.png. If I had 2 apples and 3 oranges in a bag, the probability of me drawing an orange out of a bag would be 3/5, since there are 3 oranges out of 5 possible fruits in the bag.

Keep that in mind while I list relevant #s for the enemy card equation. In map E4, there are:
 * 50 cards in total
 * 20 nodes (n)
 * 16 enemy cards (4 of each type of enemy)
 * 26 bead cards
 * 8 misc. cards (kaika, bomb, trap, etc.)

Forget about the # of nodes first. If I had just the 50 event cards, the probability of drawing an enemy card would be https://i.gyazo.com/15dadd63ceef5d6cc16aeb9670774a50.png, since there are 16 enemy cards out of 50 possible cards in the deck.

What about drawing 2 consecutive enemy cards? Recall that each card can be only drawn once, which means we have to remove one card from the deck every time we draw a card. The 2nd enemy card we draw would have the probability of https://i.gyazo.com/1b1285d231db3ce164c5bdc90f6280a0.png ,as we have 15 enemy cards left out of a total of 49 cards. Multiplying https://i.gyazo.com/1b1285d231db3ce164c5bdc90f6280a0.png with https://i.gyazo.com/15dadd63ceef5d6cc16aeb9670774a50.png, we get the probability of drawing 2 consecutive enemy cards, which is https://i.gyazo.com/27ba3d137881e4921485f5f5deaaef72.png , or roughy 9.7%.

We could use the same method to find the probability of drawing 2 consecutive enemy (let's call this enemy A) cards of the same type: https://i.gyazo.com/048a8ec2f025fb301dbf4ce3cf4a5c5e.png = 0.048%.

Now we have to consider the other cards in the deck. There are 34 other cards aside from the enemy cards. Just so we have more than 1 type of enemy, put back one card from each of the other enemies (B, C, D) in. We now have 37 cards.

The big capital P you see in the first equation at the very top stands for permutations. Using permutations, we can find the number of ways these 37 cards can be arranged regardless of order. We can also find the ways to arrange a chosen number of these cards. https://i.gyazo.com/5215e3ce5edc83cb1fb71baf16a8650f.png

This permutation you see here calculates the ways you can arrange the number of cards from the number of nodes (N) out from a pool of 37 cards.

This permutation https://i.gyazo.com/c7fc63519a19830f0c9b1187c26df42f.png calculates the arrangement of 50 cards out of N nodes.

Combining the above 2 permutations with the probability of drawing 2 consecutive enemy cards, we get an expression that shows the probability of an enemy appearing on 1 node of the map.

To find the average for the entire map, I would have to add the probability of every single node together. https://i.gyazo.com/dd6e1a31b1556a9f5f5377d008d4a8dc.png

Multiply it by 4 to get all the enemy cards in, subtract from 1 and multiply by 100 to get the percentage of 69%.

lmao... I hope someone actually read this far... If you're confused/want a more in depth explanation of parts of the equation, please reply to this post :) I skimmed over a fair bit anyways

Thanks for reading and good luck to those who haven't gotten Monoyoshi yet! 