`http://stattrek.com/m/online-calculator/hypergeometric.aspx`

I've forgotten how it works some, so I'll need refreshing, but here's an example

Population size 30

Number of successes in population 2

Sample size 4

Number of successes in sample (x) 1

Hypergeometric Probability: P(X = 1) 0.23908046

Cumulative Probability: P(X < 1) 0.747126437

Cumulative Probability: P(X <= 1) 0.986206897

Cumulative Probability: P(X > 1) 0.013793103

Cumulative Probability: P(X >= 1) 0.252873563

For exmaple, you have a pop size of 30, your deck.

of the population, 2 are considered succesive cards

your sample size, depending on if you go first or second will vary

but you'll go first and have 4 cards when your turn starts

you only need 1 of those 4 cards to be a success

- -- --- -- - -- --- -- - -- --- -- - -- --- -- - -- --- -- - -- --- -- - -- --- -- -

so the Hypergeometric Probability: P(X = 1) of 1 of those 4 cards being a success is 0.23908046 or 23.9%. Then other probabilities are listed such as X < 1, or that card not showing up at all is 74.7%, and so forth. Now you can get a lot more complicated with this. You can play out the first 5 cards of your game like probability of getting that card on your second turn and going first. Look at it from the other side as if you went second and had 5 cards.

or even try it as if you need 3 exact cards in your starting hand to high roll and dominate. let's do that.

Population size 30

Number of successes in population 3

Sample size 4

Number of successes in sample (x) 3

Hypergeometric Probability: P(X = 3) 0.000985222

Cumulative Probability: P(X < 3) 0.999014778

Cumulative Probability: P(X < 3) 1

Cumulative Probability: P(X > 3) 0

Cumulative Probability: P(X > 3) 0.000985222

Calculate

The probability of getting 3 exact cards, if I'm using this right, in a starting hand of 4 cards is 0.098%. That's less than 1%. now you might

*feel*you're seeing that more than 1% of the time, but that's the beauty of probability. It doesn't have to follow the numbers. Not exactly. Over hundreds of thousands of attempts, you might eventually see those numbers reach expectations, but that's beyond the normal scope of one Hearthstone player. Of course there are sites that culmlminate such data, I bet, so check there for more information

if you go first, don't see the 1 card you need and muligan all 3 cards for 3 more, the sample size should just be 6 then, right?

Population size

30

Number of successes in population

1

Sample size

6

Number of successes in sample (x)

1

Hypergeometric Probability: P(X = 1)

0.2

so there'd be a 20% chance to get 1 card out of 30 if you muligan your 3 starting cards for 3 more, and if you check the 7th card at the start of your turn, it's 23.3%, but if you had 2 of that card, it'd be 37% at the start of your turn.

- -- --- -- - -- --- -- - -- --- -- - -- --- -- - -- --- -- - -- --- -- -

11/25/2017 06:15 AMPosted by sky3union@Lykos

Nice. But those calculations seems to not put mulligan rule in HS into consideration. Understandable, as that number cruncher wasn't meant for it. Also, second calculation seems to not include duplicate rule.

it is as if I sensed your coming and my past shortcoming when portraying this on the forums many months ago. Hopefully what I have above solves that for everyone.

30

Number of successes in population

3

perfect 3 card draw

Sample size

9

player 2, mulligan all 4 cards, start of player 2 turn

Number of successes in sample (x)

3

getting al 3 success

Hypergeometric Probability: P(X = 3)

0.0206896551724138

or 2.1%

Of course, a good deck may have multiple sets of 3 perfect draws to increase their odds. I don't know how you stack up 2% over and over with so many overlapping card combinations.

I am on the next page of studies which may answer this for us.

`http://stattrek.com/probability-distributions/multinomial.aspx?tutorial=stat`

A binomial experiment covers card replacement. Multinomial covers 2 or more replacements, but if that formula can be suited to Hearthstone, it'd help advance everyone's understanding of card draw.

actually skipping to

`http://stattrek.com/probability-distributions/binomial.aspx?Tutorial=Stat`

.

------------------

Nah, your first example/explanation was perfect. I could figure what you meant and what each part does.

More close to,

30 (number of cards in deck)

2 (I take it you used this to mean duplicate?)

7 (as 3 cards mulligan stage +3 opening hand +1 draw)

1 (the card you want to draw in turn 1)

= 37%

but litttle higher in the actual game as, in opening hand draw (no.4-6 draw), the population is 27. And to think I used to do this manually (41.2% is the actual number btw).

Second one is just complicated : /

I would go 15 / 3 / 7 / 3 = 7%, but in actual practice it should be lower, as not all cards are duplicates in most decks. But this is a scenario when someone is looking for exact 3 specific cards, which in most case, are not really specific as what players want become different depending on opponent's play.

1. You draw 3 cards out of 30 (mulligan stage).

2. You mulligan all 3.

3. These 3 don't go back into the deck. You draw 3 from 27 left cards (opening hand). The cards from (2) is put back into the deck.

4. Turn 1. Draw (out of 27 left cards ← this part is calculated in the site you linked obviously)

starting hand of 3 cards, mullian 3, and only 1 card in your deck is the one you need.

Binomial Probability: P(X = 1)

0.18

`http://stattrek.com/online-calculator/hypergeometric.aspx`

`http://stattrek.com/online-calculator/binomial.aspx`

This one may come in handy if anyone can wield it correctly.

`http://stattrek.com/online-calculator/multinomial.aspx`

but I go for breakfast instead.

edit

11/25/2017 06:46 AMPosted by sky3unionLittle explanation of how mulligan works in HS.

1. You draw 3 cards out of 30 (mulligan stage).

2. You mulligan all 3.

3. These 3 don't go back into the deck. You draw 3 from 27 left cards (opening hand). The cards from (2) is put back into the deck.

4. Turn 1. Draw (out of 27 left cards ← this part is calculated in the site you linked obviously)

WAIT?? REALLY? :O

...

that changes everything! Thanks a ton! Still going to breakfast though but I'm coming back to test this new data out.

11/25/2017 06:41 AMPosted by sky3unionedit: what the heck? I think we're writing post in between our thought process.

I edit posts to provide updates that may not warrant new post, hopefully in a short enough period as to update before someone else posts. And yes, I sometimes use 2 successes as in cases of duplicates. But I think it can apply to non-dups too. I'd like to do it by hand or have a personal spreadsheet that uses this guy's work, but it's hard finding out how he did it all.

Now when it comes to making further draws from your deck, I don't really have to use a population of 30-x based upon how many cards I already have unless I want to do a new probability check, right? I remember doing such things though last time such as asking for P of getting 1 card out of 27 or 26. It seems it's simply a matter of looking at the whole test or a part of it as a new test.

11/25/2017 07:21 AMPosted by Boreas11/25/2017 04:21 AMPosted by ecueDThey were removed, by Blizzard or one of there payed moderators to prevent the truth from being discussed.

am i suppossed to believe this ?

truth be told, he dosen't even know for a fact what he's saying, so the likelyhood of you believing his conjecture is even less. Wild conspiracy posts that don't add anything useful to the forums aren't going to make it far, or at least shouldn't.

11/25/2017 06:40 AMPosted by LykosUnfortunately, my mulligan idea falls short. If you mulligan all 3 cards of a starting hand, you're essentially starting a whole new test. How do you combine the results of two tests then to get the combined probability.

Those mulliganed cards now go out of possible draw on replacement cards.

you draw 3, leaving deck at 27 cards.

you mulligan those 3, they are set aside and you draw 3 cards out of 27 cards.

if your going 2nd, you have to redo the math for 4 out of 30 then 4 out of 26.

your 1st draw on 1st turn is now drawing out of ALL cards left in the deck.

either 27 or 26 depending if you had coin/extra card.

But bottom line, is OP is Salty because RNG/luck doesnt favor him often enough to let him win more often.

And perception comes into play as well.

take 10 games you play and most players will remember the games they lost more than the ones they won.

They specifically remember where emotional attachment comes into play.

they were winning but opponent drew the ONLY card that could save them, top-decked for victory.

Or remembering a specific win same scenario but reversed. "i was going to lose, but i top decked, the perfect removal/burst dmg and won"

Without running a deck tracker, i know from experience, feeling like i had played only 10-20 games, but then look at my counters and see i played 35-40 already.

out of all those games, i will remember only the ones that stood out, by exciting wins, or aggravating losses.

Most times i even realize as or after i made abad play, that it was done wrong or just wasnt the right call.

Many others who blame RNG or "rigged" mechanics, dont notice their misplays, so they feel "i did everything right and still lost"

Or they dont realize they ranked up as high as they are able within their skill. because now they face equally skilled or better opponents more often.

11/25/2017 12:33 PMPosted by Alaskaneaglewhen pondering the mulligan remember that you can draw the same card again (happens quite often to me) so the probability is the same for the mulligan as the original. It's just a redraw, not a draw with improved odds.

was it because you were running two of a particular card?

`https://hearthstone.gamepedia.com/Mulligan`

If only one copy of the card is in the deck, it is impossible to draw it as a replacement.

11/25/2017 12:33 PMPosted by Alaskaneaglewhen pondering the mulligan remember that you can draw the same card again (happens quite often to me) so the probability is the same for the mulligan as the original. It's just a redraw, not a draw with improved odds.

you can draw a 2nd copy of it, but not the exact one you had during mulligan phase. however you CAN draw that original card again on 1st turn draw.

Patches is a prime example. if you mulligan him, you will never get him again as the replacement cards, but its possible to draw him turn 1 first draw.

Nov 26
-1

11/25/2017 05:55 AMPosted by Lykos

The probability of getting 3 exact cards, if I'm using this right, in a starting hand of 4 cards is 0.098%. That's less than 1%. now you mightfeelyou're seeing that more than 1% of the time, but that's the beauty of probability.

HS players have large samples. Cue up any Twith HS stream, you can see how these 1/10th of a percent opening hands are frequent and abundant.

It's not a mystery that this game gives you a 4 cost card on turn 4, 6 on 6 and so on.

What is a mystery is why Blizzard pre-programmed this in the first place.

Nov 26
-1

[quote="207578190467"]

Patches is a prime example. if you mulligan him, you will never get him again as the replacement cards, but its possible to draw him turn 1 first draw.

Mulligan'ing Patches draws patches all the time.

11/25/2017 04:21 AMPosted by ecueDMade several posts over the last 2 days about Blizzard Bots that cherry pick cards to beat you, cheating, and encouraging the rage buy. They were removed, by Blizzard or one of there payed moderators to prevent the truth from being discussed. If you stretched my aught 1mm for every time I was hacked in this game, my buttocks would bare a sign reading Lincoln Tunnel. Run delete that.

Your posts were not removed due to the content. It was either due to the way you presented them, such as using words not allowed on the forums or threads being hidden due to large amounts of down votes.

If you wish for your posts to remain visible and be something discussed, may I offer the suggestion of presenting your conversations in a manner in which you can be taken seriously and not naming your posts things like "Forum Trolls. Please downvote this".

Lastly, should you wish to continue the conversation of "things are rigged!", please refer to the main thread in which this topic is being discussed. You can find that here: https://us.battle.net/forums/en/hearthstone/topic/20759485976

Please keep any and all comments in that thread constructive and respectful. Many in there have been working hard to have a conversation without resorting to negative means so we would like to keep it that way so they have a place to express their opinions.

Thank you.