On Monday night, someone (I believe vanilla) posed a question about probabilites having to do with an Uno deck. During the break, I took the time to write a program to simulate it and find an approximate answer. Well, the thread is gone (or at least I can’t find it), but I went through all this trouble, so here’s the answer, darn it! My programming skills are not excellent, so it’s possible there’s an error in my final numbers. They at least seem reasonable, though.
The question involved playing a little game. Go through an Uno deck turning up one card a time, each time guessing what the card will be. See how many times you get it exactly right. Specifically, is 4 good? Is 6 good? How much before we suspect someone of psychic ability?
An Uno deck, by the way, consists of 108 cards. There are 52 cards of which there are two each, one card of which there are three (Wild) and one card of which there is one (Wild Draw Four).
The first strategy I devised involved just guessing, without thinking about what cards had already shown up. You could do this with a second Uno deck, by randomly drawing a card every time and guessing whatever card you draw, then putting the card back in. If the card you drew (in your second deck) matches the card that turns up (in the main deck), you get it right. If you followed this strategy, you’d average around 2.05 cards correct. You would get 4 or better 14.7% of the time, and you’d get 6 or better 2.7% of the time.
The second strategy is considerably better. With this one, every time a card comes up on the first deck, you remove that card from your second deck, and you continue guessing by drawing from your second deck. With this strategy, you’d average around 6.27 correct. You’d have a 91.6% chance of 4 or better, and a 62.1% chance of 6 or better.
I then realized there’s yet a better strategy. Remove cards as you did with the second strategy, but always make sure you guess something that there’s the most of. For instance, in the beginning there are more Wilds than anything else, so guess that until a Wild shows up. After that, make sure you guess something that there are two of remaining, until there are no longer two of anything remaining. Then, it doesn’t matter what you guess, as long as there is still one in the deck. (This strategy would be impossible with regular playing cards, because there’s only one of everything.) Average value: 8.89. Probability of 4 or better: 100%. Probability of 6 or better: 97.1%.
Now, I imagine that most people would not be able to count cards well enough to employ even the second strategy without some practice, so 4 or 6 is not bad. However, the point is that anyone can employ the third strategy if they have a good enough memory, and are practiced in it. You wouldn’t suspect anyone of psychic ability unless they consistently got above 9. And of course, how clairvoyant do you look if you keep guessing Wild until one shows up?