Probability and statistics blog

Last night, while playing Boggle, we ended up with a board without a single vowel. Not even a “Y” or “Qu”. This seemed fairly unusual, so I wondered what the chances were of such an occurrence. I found an online list of the letters each die has, and I could have written down the number of vowels on each one by hand, but whenever possible I like to do things by computer. So I fired up Hal and asked for some help with the calculations.

Apparently some European boards use a 5 x 5 grid, but here in the land of the Maple leaf our board has 16 cubes. Here are the letters on them, as I coded them into R:

d1 = c('S','R','E','L','A','C')
d2 = c('D','P','A','C','E','M')
d3 = c('Qu','B','A','O','J','M')
d4 = c('D','U','T','O','K','N')
d5 = c('O','M','H ','R','S','A')
d6 = c('E','I','F','E','H','Y')
d7 = c('B','R','I','F','O','X')
d8 = c('R','L','U','W','I','G')
d9 = c('N','S','O','W','E','D')
d10 = c('Y ','L','I','B','A','T')
d11 = c('T','N','I','G','E','V')
d12 = c('T','A','C','I','T','O')
d13 = c('P','S','U','T','E','L')
d14 = c('E','P','I','S','H ','N')
d15 = c('Y','K','U','L','E','G')
d16 = c('N','Z','E','V','A','D')

So now I had to check how many vowels were on each die. Here’s the code I used for this:

vowels = c('A','E','I','O','U','Qu','y')
vowelsFound = rep(0,16)
for(i in 1:16) {
	found = 0
	die = eval(parse(text=paste("d",i,collapse="",sep="")))
	for(l in die) {
		# Check to see if this letter is in the vowels vector
		if(l %in% vowels) {
			found = found + 1
		}
	}
	vowelsFound[i] = found
}

# Probabilities of getting a vowel for each die
pVowels = vowelsFound/6

# Probability of getting no vowel for each die
pNoVowels = 1 - pVowels

# Chance that we will get not a single vowel, including "y" and "Qu"
print(prod(pNoVowels))

If you run the code above, you should see that the probability of getting no vowels (including “Y” and “Qu”) is 0.000642. That works out to one in every 1557 boards. So it’s quite rare, but by no means so extraordinary that it crosses the Universal probability bound. Also, it’s not enough to just calculate how rare your event is, or how rare any similar or more extreme event is, and then be astounded. You also have to include all the other possible events that would have left you amazed. What about getting all vowels (much more rare)? What about getting 8 or 9 E’s, or a row or column of all A’s or E’s? It’s likely that if you add up all probabilities of all the rare events which might leave you amazed, you’ll end up with a good chance of amazement every time.

I could have stopped here, but having coded the dice, I decided to create a simplified version of the game in R. If I have a chance over the next few days I’ll add some more features.

# You will need to download a dictionary file. I found one here:
# http://svn.pietdepsi.com/repos/projects/zyzzyva/trunk/data/words/north-american/owl2-lwl.txt
words = read.table("wordlistData.dat", colClasses = "character")
words = unlist(words[,1])

# Create a random board. Plot it.
board = diag(4)
dice = sample(1:16,16)
cntr = 4
for(i in dice) {
	die = eval(parse(text=paste("d",i,collapse="",sep="")))
	board[floor(cntr/4), (cntr %% 4) + 1] = sample(die,1)
	cntr = cntr + 1
}

plot(0,0,xlim=c(0,4),ylim=c(0,4),col="white",ann=FALSE, xaxt="n", yaxt="n" )

for(m in 1:4) {
	for(n in 1:4) {
		text(m-.5,n-.5,labels=board[m,n],cex=2.75,col="#000099")
		# Draw a square the easy way
		points(m-.5,n-.5,pch=0,cex=10,lwd=1.5,col="gray")
	}
}

# How many seconds to give for each round
gameTime = 180

START_TIME = proc.time()[3]	
elapsed = 0

# Simple scoring, with 1 point per letter. 
# Dictionary only has words length 3 or longer
score = 0

cat("Find words. Hit enter after each word.\n")
while(elapsed < gameTime) {
	myWord = scan(n=1, what=character()) # Get a single word
	elapsed = signif(proc.time()[3] - START_TIME, digits=4)
	if (length(myWord)==0) {
		cat("You have", gameTime - elapsed, "seconds left. Keep going!\n")
	} else {
		
		if(elapsed < gameTime) {
			# Check if it's a real word, see if it is in dictionary
			# Convert their guess to uppercase letter
			myWord = toupper(myWord)
			
			# If it is in the dictionary, give them points
			if(myWord %in% words) {
				# Make sure they haven't used this word before TODO
			
				# Add it to their score
				score = score + nchar(myWord)
				cat("Congratulations. You are up to", score, "points.")
				cat("You have", gameTime - elapsed, "seconds left. Keep going!\n")
			} else {
				# If it isn't in the dictionary, let the user know that they got it wrong.
				cat("Sorry, that is not in the dictionary. Keep trying!\n")
			}
			
			
		}
	}
} 

cat("Out of time! ")
cat("Your final score was:", score, "points.")

Enjoy the game. Let me know if you notice any issues or have suggestions!