My kids were playing with marble mazes today. My four year old, A, decided that one marble was a mouse, running through the marble maze. Thinking about the board game “Mousetrap” he wanted a “fat cat” place on the marble maze to send the marble back to the beginning. The plastic marble maze set he was using had a piece that functions as a sort of plinko mechanism (also known as a bean machine), where the marbles bounce off of different sticks going different directions before regrouping to move onto the next piece of the maze, so I stretched a piece of tape in the gap between two of the sticks and we tried to see how many “mice” we could catch. Then we started to move the tape around to figure out which spot would be the best hunting spot. We tried each spot with 10 mice and recorded our results.
By now my seven year old, M, was interested and we drew out a little diagram and tried to calculate the probability of the mouse/marble going through each path. My son wanted the bottom half of the diagram to be symmetrical with the top half and was surprised to find that it wasn’t. I had him figure out all the answers. I offered that if he wanted we could get out pennies and move stacks of coins along the paths, half each direction, but he turned that down as too much hard work. Instead he used a calculator, double checking that each line added up to 100% and trying to understand the logic of it.
So this evening I drew out another board. I used the same style but made the widest point have seven passages ways through it. If you flip a coin and move left for heads, right for tails, it would take six identical flips to get into the outer reaches of the widest point and the likelihood of doing that is 1.5625% or 0.5 to the power of 6 (0.5*0.5*0.5*0.5*0.5*0.5). After the widest point, the outer numbers start gro wing, as the marker or marble cannot move off the edge of the page and 100% of the marbles on the outer edge move to the one that is diagnally lower than it so that by the third row from the bottom, the outer edge is actually a more likely place than the middle channel. I found it interesting that the third from the bottom row was not the same as the third from the bottom row on the smaller board.
I tease my husband that if I were a real math geek I wouldn’t be sitting here with a calculator figuring out all the rows, I would be writing the computer program to do it for me, and I wish I were doing that because I do think it would be fun to see more about how the number of rows affects the bottom section, where the channel gets smaller again.
My children, I’m sure, would not be so interested in seeing twenty different charts like this. It was good practice getting them to experiment and to do the math. Even with M using the calculator he had to think about what numbers he had to enter and why, and where his errors were when he had errors.
I might be able to get M to play a game using a large chart like this if I frame it in a story. Perhaps I’ll ask him to figure out all the ways a particular mouse could get into a particular room and then show him how his answers relate to Pascal’s Triangle.
Or I might find a story that will challenge him to compare the probability of one of these charts with possibilities in combinations…. if you had three random choices to make, say between two different shirts, two different pairs of pants and two different pairs of socks.
Each blue dot represents a choice between two paths that could be taken, but in this case there is no over lap. Each final combination can be reached by only one path and at each level any particular combination (shirt 1, pants 2 and hat 1, for example) is just as likely as any other (shirt 1, pants 1, and hat 2).
I might ask, if he was a cat hunting mice in a collection of tunnels, which of the styles would he want to be hunting in and why?