What Are the Odds to “Draw 2?” Part 2

Part 2.  The Extension

The next questions was, how do these results vary from one game to the next?

With questions like these, it’s often useful to explore the extremes first.

Extreme #1.

Imagine a deck which only contains one special.  In this case, playing a special means that none remain and you have 0% probability of holding another.  Not very useful but worth noting.  We’ll bring this up again later.

Extreme #2.

Imagine a deck of seven cards which contains two specials in which you are dealt five cards.  If you play a special, you draw the rest of the deck and thereby guarantee that you’re holding the other one.  You have 100% probability of holding another.  This is actually more useful to note than Extreme #1.  Strategy in deckbuilding games like Dominion frequently revolves around building an engine that maximizes your chances of drawing all of your deck every turn.

So we have one extreme case in which the odds drop to 0%, and another extreme case in which the odds jump to 100%. What’s going on in the middle.  That’s what I set out to discover next.

The Restrictions.

To avoid ridiculous cases like Extreme #1 or Extreme #1, let’s set out a few assumptions.

First, we have a deck of size N and your starting hand size is n.  This deck contains a quantity k of special “draw 2 cards” cards.  To avoid the ridiculous extremes, let’s agree that the deck contains at least two specials and that the deck is large enough that after playing the special and drawing the extra two cards, there will be at least one card remaining.

Put mathematically, we have that

Probability Part 3 Equations 1

How much do the odds change when your opponent plays a special card and thereby draw two cards?  The quantity of specials which could be in her hand decreases by one.  The quantity of cards you have not seen decreases by one.  The quantity of cards in her hand increases by one.

The Formula.

The change in probability is given by these formulas:

Probability Part 2 Equations 2

Example:

The deck has 60 cards.  10 of these cards are “specials.”  Players are dealt 6 cards to begin.  When a player uses one of these specials, the odds drop by approximately 6%.

Probability Part 2 Equations 3

Example:

You are playing Dominion. Your deck has 25 cards.  10 of these cards have “Draw+2.”  Each round starts with 5 cards inn hand.  When you play one of these cards, the odds you are holding another drop by approximately 3%.

Probability Part 2 Equations 4

What do you think of “draw 2 cards” cards?  Have you incorporated them into your designs?  What did you learned from the experience? Do they improve a design or weaken it?  Share with your fellow readers in the comments below.  And if you’re enjoying what you’re reading, create an account with WordPress and follow this blog.  If you keep reading, I’ll keep writing.

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