What Are the Odds to “Draw 2?” Part 1

Part 1.  The Question

The principal purpose of this blog is to bring you living debates from game design and our solutions to them.  An interesting one occurred recently which drove me on a bit of a mathematical quest.  Having taken the computational plunge, it’s now time to share with you fine folks my results.

Draw 2So it happened that we were playtesting a card game.  Luther on his turn played a card we’ve seen in a huge number of games; he played the special “draw 2 cards” card.  So far, so good, right?  We’ve seen this card many times in many different games.  Our postgame discussion led to a debate over whether playing this special meant that he was more likely or less likely to be holding another.

We began to explore the possibilities.  Luther holds more cards, that increases the odds but he also played one of these specials and that reduces the odds. It was hard to tell from the outset.

We did the calculations.  This particular game has a 99 card deck and 9 of these cards are the “draw 2 cards” special.  Players are dealt an 11 card hand.

The easiest way to determine the odds that Luther is holding a special is to calculate the odds that he is not and then subtract this result from 1.  The probability that he is not holding a special in his first eleven cards is

Probability Part 1 Equations 1

In the interest of time and space, I elected to assume that you are already familiar with the mathematics of permutations. If you find yourself wanting a refresher, there is a solid description at Math is Fun.

Subtracting this from 1 gives us

Probability Part 1 Equations 2

So there was roughly a 67% chance that he received a special in his opening deal.

Now he plays a special, which reduces his hand by one card and draws two cards.  He is now holding twelve cards but there is one fewer special in the available supply since he just played one of them.  The odds that he is not holding a special in these twelve cards is

Probability Part 1 Equations 3

Subtracting this from 1 gives us

Probability Part 1 Equations 4

There is now roughly a 71% chance that he holds a special.  We have our result.  Not only did the odds go up, the odds rose by nearly 4%.

But this is not the end of the story.

It’s nice to know that the odds go up by 4% in this particular game but what about all the other games with similar conditions?  What about later in this same game, once a few cards have been seen?  Is there some point at which the odds slip?  When is that?  These questions propelled me to dig deeper.  Our next installment will seek to answer these questions.

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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