The ORIGINAL Plinko Board
We turn our attention to the original Plinko board which has 9 columns and 13 rows.
One way to quantify the likelihood of winning certain amounts is to count the paths from each of the top slots to the ending slots. To do this, we can use the same Pascal's Triangle technique that we used for the smaller board.
To obtain the number of paths to each of the slots using Pascal's Triangle, we display Pascal's Triangle superimposed on the big Plinko game board. To account for the loss of paths when a chip comes into contact with the side of the board, we must subtract. The subtractions we must make are shown in RED on the board. The final path counts for the affected cells are shown in blue.
Thus, there are 54 paths from E to A', 219 paths from E to B', etc.
Instructor's Note: Path counts from the various starting slots are shown here.
The probabilities that a chip starting in slot E will land in a particular slot can be computed using an argument based on the number of paths and tree diagrams. Click here. However, the technique involving Pascal's Triangle that we observed for the small board case also works for the larger Plinko board (this has been verified for a Plinko board with 9 starting and ending columns with 13 rows using probability tree diagrams by S. Lanier and S. Barrs and independently using path counts by L. Roberts).
To determine the probabilities that a chip will fall from slot E to the various slots at the bottom can be computed in an approach similar to what we used for the smaller board. Superimposing Pascal's Triangle over the game board in Figure 8, we look at the last row. The sum of the entries is 212 = 4096. This is the denominator of the probability fractions because there is at least one path that has probability 1/212.
To compute the numerators for the probability fractions using the thirteenth (n = 12) row of Pascal's Triangle, reflect the entries outside the board to the inside and add.
The probabilities with which a chip will land in one of the slots at the bottom given that it is dropped from slot E are computed and shown below as fractions as well as to three decimal places.
A summary of the conditional probability computations (using the approach involving Pascal's Triangle) for a chip starting in slots A, B, C, and D is given here.
When playing a game of chance that involves money, it is interesting to think about how much money we might expect to win. In mathematical terms, this concept is defined to be expected value or mathematical expectation. We define expected value (in the context of the Plinko game) as follows. This will help us to determine expected winnings associated with playing the Plinko game.
Using asummary of the probabilities, we can now discuss probabilities for winning and the expected values, i.e. the expected winnings, associated with playing the game.
Summary of Conditional Probabilities:
What do the conditional probabilities tell us about the probabilities that we'll win money? Notice that for each chip that we use, there are two columns in which we can win $0, $100, $500, and $1000, so we add the conditional probabilities in each row corresponding to $0, $100, $500, and $1000, respectively. There is only one column in which we can win $10000. This information is summarized in the table below.
Based on the theoretical probabilities we have calculated, we can make the following observations.
We can use the probabilities above to determine the expected winnings.
For example, for each chip dropped in A,
winnings = 0.113*0 + 0.226*100+0.387*500+0.242*1000+0.032*10000
In a similar way we can compute the expected winnings for a chip dropped in the other slots. The expected winnings are summarized in the table below.
In the table above, we used the theoretical probabilities to obtain the expected winnings. If we obtain experimental probabilities, the expected value is the average value of the outcomes over many repetitions. So if we run many simulations of the Plinko game and determine the winnings of each trial, the expected value is the average of the winnings over the number of trials.
Do these numbers GUARANTEE that we would win a certain (hopefully significant) amount of money if we use the probability analysis to develop a strategy for playing Plinko on The Price is Right?
The short answer is "NO."
Probabilities only express likelihoods in which we can place more and more confidence with a larger and larger number of trials.