Math Games

Last night Natalie introduced me to a math game that she had learned at school. On the top of separate sheets of paper we each wrote “1,000.” Then we took turns rolling a die, and then subtracting the amount rolled from the thousand. The object of the game was to be the first to zero. My first thought was that this may be the most boring game ever invented. My second thought was that this would take a long time, about 285 turns each if probabilities hold. My first roll was a one, so I wrote “999” beneath the 1,000. “Dad, you have to show your work,” she explained. Are you kidding? I don’t know how to do that. It isn’t work. I can safely promise that I can subtract any integer that is between zero and seven from ANY number in my head. 23,720,385, 379 minus 5? Got it. Sorry Sister, I’m not going to borrow and “show my work.” I did suggested that we use two dice, so the game would go a little faster and we could do more complex subtraction. Natalie acquiesced. She jumped out to a quick lead as I rolled several successive 2s and 3s. Armed with a rudimentary knowledge of probabilities, I started talking some smack. I told her that her lead wouldn’t hold, and that if we played long enough, at some point we would be tied. Right? Is that true? If we played long enough? Well, she tired of the game and shortened it to just 400 points. She won handily but in a solitary moment of thought, I had to concede that my claim was not true. It may be likely that we would be tied at some point, but never a sure thing. If I begin a turn more than ten points behind (or ahead) , then there is a zero probability that after her roll we will be tied. If I am within ten, there is a chance to end up tied, but every time one of us rolls, there will always be a chance that we won’t roll the tying number. Sure, if we played for three days, chances are that at some point we’d be tied, but it is never a certainty. The longer we play, the higher the probability, but it is asymptotic, never reaching 100%. If we played for a year, the chances might be 99.9999%, but it can never reach 100% Does that make sense? The fact hat 100 previous potential tying rolls didn’t tie the game have no bearing on the current roll. If you flip a coin five times and get heads each time, there are no greater odds of getting tails on the sixth try. I knew a woman expecting her 4th, and felt that since the first 3 were girls, she had a good chance at getting a boy. No you don’t. I don’t know of any connection between gender selection and previous births. Anyway, that is where my thoughts were last night as I drifted off to sleep. Maybe the game isn’t so boring after all.