Why is it 40%?
The draw calculator says that four copies of a card in a sixty card deck gives you about a forty per cent chance of seeing one in your opening hand. Here is exactly where that number comes from, one step at a time. If you are ten and you can multiply, you can follow every line of this.
First, picture the deck
Sixty cards, shuffled. Four of them are the card you want. Every other card is a perfectly nice card that is not the one you are after right now, so we will call those the misses.
You draw seven of these for your opening hand. The question is simple to ask and surprisingly easy to get wrong: what are the chances that at least one of those seven is gold?
Have a guess now, before you read on. Write it down. Guessing first is the part that makes the answer stick.
Two answers that feel right and are not
Almost everybody tries one of these two first, and it is worth seeing why they do not work, because the reason is the whole idea.
Guess one
6.7%
Four cards out of sixty. That is a correct answer to a different question: the chance that one single card off the top is gold. You are drawing seven, not one.
Guess two
46.7%
Seven times the first guess. Closer, and still wrong, for two reasons worth understanding properly.
The first reason: when you add up chances like that, you count some hands twice. A hand that holds two gold cards gets counted once for each of them, so the total comes out too big.
The second reason: the deck changes as you draw. Once you have taken a card, it is gone. The second pick is out of fifty-nine cards, not sixty. Multiplying by seven pretends the deck stays the same all the way through, and it does not.
That second reason has a name. Mathematicians call it drawing without replacement, which just means you do not put the card back.
The trick: count the misses instead
Here is the move that makes this easy, and it is a trick worth keeping for the rest of your life.
Working out the chance of getting at least one gold card is fiddly, because at least one means one, or two, or three, or four, and you would have to work out all of those and add them up.
But there is only one way to get none at all: every single card you draw has to be a miss. That is much easier to count. So we work out the chance of getting none, and then take it away from 100%.
chance of at least one = 100% − chance of none
If there is a 60% chance of missing completely, then there must be a 40% chance of not missing completely. Those are the only two things that can happen, so they have to add up to everything.
Seven picks, and the deck shrinks each time
So: what is the chance that all seven cards are misses? We go one pick at a time. There are 56 misses in the deck at the start.
Pick 1
56/60
56 misses left,
out of 60 cards
Pick 2
55/59
55 misses left,
out of 59 cards
Pick 3
54/58
54 misses left,
out of 58 cards
Pick 4
53/57
53 misses left,
out of 57 cards
Pick 5
52/56
52 misses left,
out of 56 cards
Pick 6
51/55
51 misses left,
out of 55 cards
Pick 7
50/54
50 misses left,
out of 54 cards
Look at what happens down the line. The top number drops by one each time, because you have taken a miss out of the deck. The bottom number drops by one too, because the deck is one card smaller. That is the shrinking deck, written down.
Now multiply all seven fractions together. Multiplying is right here because we want all seven things to happen, one after another.
56/60 × 55/59 × 54/58 × 53/57 × 52/56 × 51/55 × 50/54
= 60.05% chance of drawing no gold card at all
And so, taking that away from 100%
39.95%
chance of seeing at least one in your opening seven
How close was your guess?
What the grown-up formula says
If you look this up, you will find something that looks far scarier than what we just did. It is called the hypergeometric formula, and for our question it looks like this.
C(56, 7) ÷ C(60, 7)
C(56, 7) is read as “56 choose 7”. It means: how many different groups of 7 cards could you make from 56 cards.
So the top is the number of seven-card hands you could build using only misses, and the bottom is the number of seven-card hands you could build out of the whole deck. One divided by the other is the fraction of all possible hands that contain no gold card at all.
Which is exactly the thing we worked out the slow way. And it gives exactly the same number: 60.05%.
That is the honest secret about a lot of scary looking maths. It is usually a short way of writing something sensible that you could have worked out yourself with more time and more paper.
The bars on the calculator
The tool also shows you the chance of drawing exactly none, exactly one, exactly two, and so on. Each one is worked out the same way, and they have to add up to 100%, because one of them is definitely going to happen.
Notice how quickly it falls away. Drawing one copy in your opening hand is common. Drawing three is something you will wait a very long time for. This is why players put search cards in their decks: it is far easier to find a card on purpose than to wait for it to turn up.
The mulligan is the same sum again
If your opening hand has no Basic Pokémon, you have to show it and draw a fresh seven. That is a mulligan. Working out how often it happens is the identical question with different numbers.
With 12 Basic Pokémon in a 60 card deck, there are 48 cards that are not Basics. What is the chance that all seven of your cards come from those 48? Same trick, same shrinking deck.
19.1%
about one game in 5
This is the number that should change how a deck is built. Add more Basic Pokémon and it drops. That is not an opinion about the game, it is arithmetic, and your child can prove it to you in about a minute with the calculator.
Three things to remember
- When a question asks for at least one, work out the chance of none and take it off 100%. It is nearly always the shorter road.
- You cannot just add chances up. Adding counts some outcomes twice, which is why the seven times answer came out too big.
- If you do not put things back, the numbers change with every pick. Watch the bottom of the fraction get smaller.
Try this one yourself
A deck has 60 cards and 4 copies of your card. You have drawn your opening seven and taken three more turns, so you have seen ten cards. Work out the chance of having seen at least one, using the trick from step three. Then put the numbers into the calculator and see whether you were right.
Hint: your chain of fractions will be ten long instead of seven, and it starts in exactly the same place.