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.

 A B C D E F G H I 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 -1 9 45 120 210 252 210 120 45 10 -1 9 1 1 11 55 -1 54 165 330 462 462 330 165 55 -1 54 11 1 1 12 66 -12 54 220 -1 219 495 792 924 792 495 220 -1 219 66 -12 54 12 1 A' B' C' D' E' F' G' H' I'

Thus, there are 54 paths from E to A', 219 paths from E to B', etc.

Activity:  This can be done in groups.  Using the Pascal's Triangle approach, count the paths from slots A, B, C, and D.  Use a symmetry argument to get the number of paths from F, G, H, and I.   Complete the table below.

 TO FROM A' B' C' D' E' F' G' H' I' A B C D E 54 219 475 792 924 792 495 219 54 F G H I Prize \$100 \$500 \$1000 \$0 \$10000 \$0 \$1000 \$500 \$100

Based on the results of the path counts, if you have only one chip to play, into which slot would you place it?  Explain your answer and be sure to consider the possibility of getting \$0.

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.

 A B C D E F G H I 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 220 66 12 1 A' B' C' D' E' F' G' H' I' \$ 1 0 0 \$ 5 0 0 \$ 1 0 0 0 \$ 0 \$ 1 0 0 0 0 \$ 0 \$ 1 0 0 0 \$ 5 0 0 \$ 1 0 0

Figure 8.  Pascal's Triangle superimposed on larger Plinko Board.

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.

 P(A'|E) P(B'|E) P(C'|E) P(D'|E) P(E'|E) P(F'|E) P(G'|E) P(H'|E) P(I'|E) 0.016 0.057 0.121 0.193 0.226 0.193 0.121 0.057 0.016

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.

Expected Winnings

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.

 Suppose E1, E2,  ...,En.  are disjoint, exhaustive events (that is, at least one of them must happen in an experiment) with probability p1, p2, ..., pn .  Suppose that the events are associated respectively with amounts of money m1, m2, ..., mn.  Then the mathematical expectation (expected value) for this situation is defined to be  .

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:

 Column End Begin A' \$100 B' \$500 C' \$1000 D' \$0 E' \$10000 F' \$0 G' \$1000 H' \$500 I' \$100 A 0.226 0.387 0.242 0.107 0.032 0.006 0.000 0 0 B 0.193 0.346 0.247 0.137 0.057 0.016 0.003 0.000 0 C 0.121 0.247 0.241 0.196 0.121 0.054 0.016 0.003 0.000 D 0.054 0.137 0.196 0.226 0.193 0.121 0.054 0.016 0.003 E 0.016 0.057 0.121 0.193 0.226 0.193 0.121 0.057 0.016 F 0.003 0.016 0.054 0.121 0.193 0.225 0.196 0.137 0.054 G 0.000 0.003 0.016 0.054 0.121 0.196 0.241 0.247 0.121 H 0 0.000 0.003 0.016 0.057 0.137 0.247 0.346 0.193 I 0 0 0.000 0.006 0.032 0.107 0.242 0.387 0.226

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.

 Winnings Begin Column \$0 \$100 \$500 \$1000 \$10000 A 0.113 0.226 0.387 0.242 0.032 B 0.153 0.193 0.347 0.250 0.057 C 0.250 0.121 0.250 0.258 0.121 D 0.347 0.057 0.153 0.250 0.193 E 0.387 0.032 0.113 0.242 0.226 F 0.347 0.057 0.153 0.250 0.193 G 0.250 0.121 0.250 0.258 0.121 H 0.153 0.193 0.347 0.250 0.057 I 0.113 0.226 0.387 0.242 0.032

Based on the theoretical probabilities we have calculated, we can make the following observations.

1. Regardless of the starting slot you choose, there is always a better chance of winning \$0 than \$10000.

2. The best chance for winning \$10000 is if you choose starting slot E.  However, this also gives the best chance for winning \$0.

3. Regardless of the starting slot you choose, the probability for winning \$1000 is 0.25.

We can use the probabilities above to determine the expected winnings.

For example, for each chip dropped in A,

expected winnings = 0.113*0 + 0.226*100+0.387*500+0.242*1000+0.032*10000
= \$778.10.

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.

 Begin Column Expected Winnings A \$778.10 B \$1012.80 C \$1605.10 D \$2262.20 E \$2561.70 F \$2262.20 G \$1605.10 H \$1012.80 I \$778.10

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.

 Some PLINKO Trivia:  As of July, 2001, the most money ever won by a contestant playing the Plinko game on The Price is Right was \$22,100 (source:  http://www.cbs.com/daytime/price/)

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?

Probabilities only express likelihoods in which we can place more and more confidence with a larger and larger number of trials.

Activity:  It is interesting to compare theoretical probabilities to the corresponding experimental probabilities for the  Plinko board.  TI-83, TI-89 programs and a Javascript routine are available to facilitate simulations for the experimental conditional probabilities as well as the probabilities associated with a random starting slot.  Click on the appropriate buttons below to open the simulation windows.  Instructions are available as the simulation page opens.  Each simulation runs 100 trials.

If you run the simulation in Internet Explorer, only the last path will be shown.
If you run the simulation in Netscape, each path will be shown (very quickly--depending on how quickly your screen refreshes, only part of the path may be displayed).

You can see and download a sample activity data collection sheet here.  (Word format).

TI calculator simulations:

 View the code and download the TI-83 Plinko program here. View the code and download the TI-89 Plinko program here. Javascript simulations: 100 trials from user-selected starting position: 100 trials from random starting column: