Pokémon TCG: Sword and Shield—Brilliant Stars

3 versus 4?

jjkkl

jjkkl

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I'm not the greatest as probability, so I'm wondering to those more familiar with this whether there is:

A) someone willing to tell me the percentage difference of getting a card in your opening hand with 3 cards versus 4 cards
B) a program / formula that'll help me with it

In a nutshell, if I want to start with a basic, what's the percentage between starting with 3 cards versus 4 cards? If I had Card A, and I wanted it in my hand in the first 7 draws, would the 1 extra card make a massive difference to warrant it in terms of probability?
 
I think it has more to do with how many of the OTHER basics you have. Because as long as you don't draw into them, you either mulligan or draw the basic you want. I've been wanting to set up some calculations, but probability was never my strongest side/most fun part of maths last year so I haven't come through to do it yet. If I ever come up with a formula I'll let you know. :)
 
Hmm, that would make sense. Would you know, however, of the difference between 3 and 4 are mitigate-worthy.

Awesome. That would be much appreciated.
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If you really just want the basic percentage of draw card X in your opening hand all you have to do is take how many copies of card X you have in your deck divided by 60. So in your case 3/60 or 4/60 which gives 5% and 6.5% respectively. So a difference of 1.5%. This means that if you want that one card you have a 1.5% better chance of it in your opening hand. But if you consider cards that all you to search that card out your percentage does go up for allowing you to get it out turn one. And to your follow up question the extra card does make a huge difference in being able to get it out. Think of it this way you take your opening hand and your prizes out of your deck to start the game and card X is not there. You are left with 47 cards in your deck following the same process as above 3/47 and 4/47 gives 6.3% and 8.5% respectively this is a huge difference. This assumes that none of your original card X's are in your prizes.
 
Gyarados11 said:
Pokepedia is an excellent site to use to figure out these types of probabilities.
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It really really is--it includes mulligan calculations, tells you what the probability of having multiple copies of a card in your hand instead of just one, lots and lots of info.
 
I can see you have been given some helpful links already, but there is an easy way to think about probabilities in drawing a card from a deck.

So calculating probabilities, 0 means the probability is 0% (not possible) and 1 means 100% probability (will always happen). I will also write probability = p(what happens according to this probability) and times = x. I will post the percentiles of my calculations, too, but these things are easier to understand as divisions.

The basic idea for one draw from a deck is a distribution: The number of wanted cards in deck (currently) PER The total number of cards in deck (currently). For example, say you have a full deck of 60 cards in front of you, and you have 3 of the card you want to pull. The probability for pulling the right one in your first draw is then 3/60.

Well, calculations are a bit more tricky when you can draw several times because
1) The amount of cards in your deck decrease with every draw
2) It is possible to get the wanted card multiple ways.

So when trying to draw 1 card from the cards you need, it is often easier to calculate the probability of NOT pulling the card you want. Then the probability of pulling the card can be written: 1 - p(not pulling the card you need).
For example, if you have a deck of 60 cards. Let's say you want to draw a pikachu and you have 3 pikachus in your deck. You draw the deck 3 times.
In your first draw, the probability of not getting a pikachu is:
(The amount of cards that are not pikachu) / (The number of all cards total) = (60-3) / 60 = 95 %
In your second draw the probability of not drawing a pikachu is (for that draw alone): (59-3) / 59 = 94.9%
In your last draw the probability of not pulling a pikachu is (for that draw alone): (58-3) / 58 = 94.8%

So the probability of not pulling a pikachu in either of the three draws becomes:
[(60-3) / 60] x [59-3) / 59] x [(58-3) / 58] = 0.95 x 0.949 x 0.948 = 0.85 = 85% (roughly).

That means during those 3 draws you will draw at least one pikachu is 1-0.85=0.15 = 15%.

I told you the calculations itself are easy, but actually quite laborious (especially if you want to calculate the probability of 1, 2 or all three pikachus ending up in the prizes first..). Hope this helped anyway, at least the way to think about these.
 
RevYolution said:
If you really just want the basic percentage of draw card X in your opening hand all you have to do is take how many copies of card X you have in your deck divided by 60. So in your case 3/60 or 4/60 which gives 5% and 6.5% respectively. So a difference of 1.5%. This means that if you want that one card you have a 1.5% better chance of it in your opening hand. But if you consider cards that all you to search that card out your percentage does go up for allowing you to get it out turn one. And to your follow up question the extra card does make a huge difference in being able to get it out. Think of it this way you take your opening hand and your prizes out of your deck to start the game and card X is not there. You are left with 47 cards in your deck following the same process as above 3/47 and 4/47 gives 6.3% and 8.5% respectively this is a huge difference. This assumes that none of your original card X's are in your prizes.
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I'm sorry, but most of what you said is incorrect. You lack too many factors. But still, you added one I hadn't though of; starting with a free retreat Pokémon and means to search for the one you want to start with. So thank you.

Blue Wave said:
@above: That's the percentage of drawing your card in ONE top deck, whereas an opening hand is seven cards.

A) At 3 cards, it's about 33%. At 4, it's 40%.

B)http://pokegym.net/forums/showthread.php?t=124908'

That's ryanvergel's article about consistency, which should have the formula you need.
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Ok, maths are hard enough in my native language to understand anything in that article. I don't know where the 40 % with 4 basics came from, but I'm pretty sure it's incorrect. The chance of drawing into the starter you want isn't entirely dependent on how many copies of that starter you play, you also have to factor in the chance of starting with other basics, mulligans and starting with free retreats and means to search for the basic you want. And with all these factors it can't possibly always be 40 % with 4 basics.
 
What about determining the odds of a trainer or supporter in your opening hand? That's pretty useful for Collector odds.
 
This 40% is solely the chance of having it in your opening hand before doing anything, it doesn't factor in mulligans, search cards or anything like that. Other basics are simply included in the other 60 cards - they have a chance of their own, and of course there is a chance of having both the card you're playing 4 of and another card in your opening hand, but I rarely go further than checking the chance of that. And as for how I got 40%:

1-(56/60)*(55/59)*(54/58)*(53/57)*(52/56)*(51/55)*(50/54) = 0,399499626/1 chance of having that basic in your opening hand of 7 cards.
It is 100% minus the probability of not drawing the card in question for each card in your opening hand, because the likelihood of drawing the card is different when you have 60 cards in your deck and 55 cards in your deck.

English isn't my native language either, but judging by the math in the article and my own calculator this should be correct.
There are programs that do this for you, but to me it's just more comfortable puling out my trusty calculator. :p

EDIT: Haunter_Fanboy's link is very useful, I had no idea about it. However, it showed that 40% indeed is the chance of getting a card you play 4 copies of in the opening hand.
 
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Blue Wave said:
This 40% is solely the chance of having it in your opening hand before doing anything, it doesn't factor in mulligans, search cards or anything like that. Other basics are simply included in the other 60 cards - they have a chance of their own, and of course there is a chance of having both the card you're playing 4 of and another card in your opening hand, but I rarely go further than checking the chance of that. And as for how I got 40%:

1-(56/60)*(55/59)*(54/58)*(53/57)*(52/56)*(51/55)*(50/54) = 0,399499626/1 chance of having that basic in your opening hand of 7 cards.
It is 100% minus the probability of not drawing the card in question for each card in your opening hand, because the likelihood of drawing the card is different when you have 60 cards in your deck and 55 cards in your deck.

English isn't my native language either, but judging by the math in the article and my own calculator this should be correct.
There are programs that do this for you, but to me it's just more comfortable puling out my trusty calculator. :p

EDIT: Haunter_Fanboy's link is very useful, I had no idea about it. However, it showed that 40% indeed is the chance of getting a card you play 4 copies of in the opening hand.
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If it doesn't factor in all those other very important factors then I don't see any use for it. You can't partly calculate the probability of something and then say that the numbers you end up with are right.
 
The exactly same formula can be used to calculate the chance of a mulligan (just replace the 56 in 56/60 and so forth with the number of your basic Pokémon which will give you the chance of having one of those in your hand. The rest of 100% of starts will be mulligans. If you run 10 basics, the chance of a mulligan is roughly 26% - after that mulligan, you once again have a 40% chance of drawing whatever card you run 4 copies of.
If you really wanted to see what the chance of getting a Reshiram, which you would run 4 copies of, in your hand T1, you would even have to factor in Pokémon Communication and Pokémon to shuffle back into your deck with that - how you do that I have no clue about.

These calculations are for the odds of starting with your Pokémon of choice, which will remain 40% every time the deck does not mulligan. The only thing other basics have to do with it is how often you mulligan and have another shot at drawing into your preferred starter. After all, if you have a deck with 60 basics, you will never mulligan, but the chance of drawing into your starter will remain the same every time.

Still, this usually tends to be enough - you commonly want to know the odds of starting your first turn with a certain basic active rather the odds of having a certain basic active by the end of T1. This entire thread was about the odds of getting a certain card in your starting hand before you even draw your first card of the game - wasn't it?
 
Blue Wave said:
The exactly same formula can be used to calculate the chance of a mulligan (just replace the 56 in 56/60 and so forth with the number of your basic Pokémon which will give you the chance of having one of those in your hand. The rest of 100% of starts will be mulligans. If you run 10 basics, the chance of a mulligan is roughly 26% - after that mulligan, you once again have a 40% chance of drawing whatever card you run 4 copies of.
If you really wanted to see what the chance of getting a Reshiram, which you would run 4 copies of, in your hand T1, you would even have to factor in Pokémon Communication and Pokémon to shuffle back into your deck with that - how you do that I have no clue about.

These calculations are for the odds of starting with your Pokémon of choice, which will remain 40% every time the deck does not mulligan. The only thing other basics have to do with it is how often you mulligan and have another shot at drawing into your preferred starter. After all, if you have a deck with 60 basics, you will never mulligan, but the chance of drawing into your starter will remain the same every time.

Still, this usually tends to be enough - you commonly want to know the odds of starting your first turn with a certain basic active rather the odds of having a certain basic active by the end of T1. This entire thread was about the odds of getting a certain card in your starting hand before you even draw your first card of the game - wasn't it?
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Yes, it was for the opening hand, but I think OP will be just as satisfied with having the preferred starter active by the end of T1, because that is the goal when you are playing. What I think you are saying is that 40 % is not completely correct if all the factors are considered - but that it's good enough? If it is, then I do not agree. Especially not when the number 40 % spreads like a disease to every player who doesn't know better, instead of giving them the real numbers. You calculations are correct, but you can't say that when you are actually playing a game you have a 40 % chance of starting with a specific basic/having it active T1 if you have 4 in your deck. The calculation is correct, but it's not the correct number when it comes down to playing the game.
 
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