An interesting application of Pascal's Triangle is in modeling a gambling game called Plinko. In Plinko, a ball is dropped down a series of nails offset from each other. The idea behind the game is that the ball can bounce either way off a nail down to the next one all of the way until it hits the bottom where it is collected in some sort of slot. Originally, the game wasn't set up in the rectangular shape shown above, but rather it was a triangle with one pin at the top and many slots at the bottom. Assuming that the game was designed perfectly, every ball dropped would have an equal chance of going either direction at each split. This results in numerous paths a ball could take to reach the bottom.
Due to the way Pascal's Triangle is generated, it turns out that Pascal's Triangle exactly models the distribution of paths on a perfect Plinko board. The idea is that each nail or slot is represented by a number in the triangle. At the top, the lone 1 illustrates the idea that the ball must hit the first nail and there is only one way to get there. The two 1's in the second row show how there is only one way to reach each of the nails in the second row. The 2 in the middle of the third row represents how there are 2 ways to reach this nail because a ball can bounce off of either the nail above it to the left or a nail above it to the right. This correlation continues down the triangle showing the number of possible paths to each split. This number of paths does not hold much meaning on its own, but it can be used to calculate the probability of landing in a given slot. To do this, we need to know the total number of paths to reach the bottom row. This could be done by simply adding all of the numbers in a row, but there is actually a faster way to do this. It turns out that the sums of every row of Pascal's Triangle are increasing powers of 2. You can determine how far down a row is and calculate the correlating power of 2 to get its sum. The probabilities of hitting a slot can be used to help making decisions while trying your luck at plinko.
1) How could Pascal's Triangle be adapted to apply to rectangular Plinko boards?
2) What factors keep most Plinko boards from being ideal (having an equal chance of bouncing either way off of a nail)?
3) How could you use your knowledge of probability to your advantage in games such as Plinko?