So you want to win Plinko?

On January 3, 1983, the newest game of The Prize is Right was introduced to the world: Plinko (CBS Interactive, n. d.) Contestants stood at the Plinko board on stage, preparing to drop their chips on the pachinko-like board for the chance to win different prizes depending on which room the chips landed in. Every time a contestant tried to drop a chip on the board, the roaring voices of the audience tried to guide them on where to drop the chips to win the most prize money. Where should the contestant drop their chips to win the most prize money? What would the voice of mathematical reason tell the contestants? Most importantly, would the contestants listen to the quiet voice of mathematical reason or the overwhelming voice of the audience? We set out to determine what it was that the quiet voice of mathematical reason had been telling the contestants all along. We built a computer model that simulated the game and determined where the chips were more likely to land depending on where they were dropped. From there, the show's producers began to ask questions about how much they might have to pay the contestants if they were realistic. As we built the computer model, we started to ask more questions about what would happen if we tweaked the board. As it turned out, the Plinko-type mini-games appeared in many locations with different dimensions than in the original game. The computer model was modified to see how the different dimensions of the board would affect the placement of the chips.

Introduction

Over 60 years ago, the world was introduced to the long-running game show The Prize is Right.[1] In 1972, the game show was relaunched into its current form and has been running ever since.[1] On January 3, 1983, the contestant who placed the closest bid on an item won the opportunity to play a new price-fixing game. In this game, they had the chance to win up to $25, 000.[3] However, as always, it's possible to win nothing. Contestants win monetary value by dropping chips from a large, slingshot-like board with nine slots, thirteen rows of pegs, offset by every other row, and nine rooms at the bottom where chips can be placed. The chips fall into nine dollar-labeled rooms at the bottom of the board, including $0. The contestant is given one chip and can win up to four chips by guessing the prices of four items. Here are some fun facts about Plinko from Prize is Right:

No one has ever put all five chips into the center chamber[3].

• To date, the most money ever won on Plinko is $39, 200[4].
• For consistency, there are only 10 Plinko chips, which are locked after use[2].
• The chips are designed to make a "plink" sound on the board.[3]
• The math behind it

The math behind the basic Plinko game is very simple and has been studied. The board itself looks like Figure 1 below.

Figure 1: A standard Plinko board.

As the chips fall down the pachinko-like board, there are two possibilities for each time the chip hits a peg. So there is some interesting math involved in the number of paths to get to the chamber across the bottom. It may be easier to work out the number of paths on a small, simple Plinko board like the one in Figure 2 below.

Figure 2: A smaller version of the Plinko board.

Figure 3 below is a tree diagram showing the probability of landing in each chamber across the bottom based on various possible paths through the board. Also, assume that the chip is dropped from slot B.

Figure 3: Paths and Probabilities of a Chip Landing in a Certain Output Slot.

With these probability, the number of routes to the bottom of the actual pingko can be easily seen. It turned out that the number of paths was related to the simple mathematical structure of the triangle of Pascal in the scene of Fig. 4.

Fig. 4: Pascal triangle

The important thing here is the line that starts with 1, 12, 66, ... This line will define the number of paths from the center slot at the top of the board to the bottom of the board, but there are only nine chambers on the bottom of the board, only nine rows. This line defines the number of paths from the center slot at the top of the board to the bottom of the board. Considering that the chips bounce back from the side, it is easy to correct this. Table 1 shows the necessary correction when the board height is 13 rows and the width is 9 rows. The addition to the 12 and 220 added to 66 is a mathematical shown by making a triangle of Pascal after the 13th row of 66 in the 13th row. This is considered to be physically folded on the left side of the triangle, which has a folded line on the vertical line between 12 and 66, on the remaining triangle. Note that 12 line up with 66 and 1 line up with 220. Similar to the right side of the triangle, fold the right side of the triangle with the remaining part of the triangle,

+12

1	12	66	220	495	792	924	792	495	220	66	12	1
+12	+1	+1	+12
78	221	495	792	924	792	495	221	78
A’	B’	C’	D’	E’	F’	G’	H’	I’

In other words, there are 78 ways to enter $ 100 (A 'and I'), how to enter the $ 500 chamber (B 'and J'), and 495 ways to enter the $ 1000 chamber (c (c)). 'And g'), there are 792 ways to enter the $ 10 chamber (d 'and e'), and 924 (e ') are on the $ 5, 000 or $ 10, 000 chamber. The difference in amount varies depending on the year of production. Again, these are the probability of dropping chips from the middle slot. Considering dropping a chip to a different slot at the top means simply moving a broken line.

Research Terms 1 < Span> With these probability, the number of routes to the bottom of the actual pinker can be easily seen. It turned out that the number of paths was related to the simple mathematical structure of the triangle of Pascal in the scene of Fig. 4.

Fig. 4: Pascal triangle

The important thing here is the line that starts with 1, 12, 66, ... This line will define the number of paths from the center slot at the top of the board to the bottom of the board, but there are only nine chambers on the bottom of the board, only nine rows. This line defines the number of paths from the center slot at the top of the board to the bottom of the board. Considering that the chips bounce back from the side, it is easy to correct this. Table 1 shows the necessary correction when the board height is 13 rows and the width is 9 rows. The addition to the 12 and 220 added to 66 is a mathematical shown by making a triangle of Pascal after the 13th row of 66 in the 13th row. This is considered to be physically folded on the left side of the triangle, which has a folded line on the vertical line between 12 and 66, on the remaining triangle. Note that 12 line up with 66 and 1 line up with 220. Similar to the right side of the triangle, fold the right side of the triangle with the remaining part of the triangle,

+12	+12	Table 1: Correction of possible paths with a pinker board	In other words, there are 78 ways to enter $ 100 (A 'and I'), how to enter the $ 500 chamber (B 'and J'), and 495 ways to enter the $ 1000 chamber (c (c)). 'And g'), there are 792 ways to enter the $ 10 chamber (d 'and e'), and 924 (e ') are on the $ 5, 000 or $ 10, 000 chamber. The difference in amount varies depending on the year of production. Again, these are the probability of dropping chips from the middle slot. Considering dropping a chip to a different slot at the top means simply moving a broken line.	Research Terms 1 With these probability, the number of routes to the bottom of the actual pingko can be easily seen. It turned out that the number of paths was related to the simple mathematical structure of the triangle of Pascal in the scene of Fig. 4.	Fig. 4: Pascal triangle	The important thing here is the line that starts with 1, 12, 66, ... This line will define the number of paths from the center slot at the top of the board to the bottom of the board, but there are only nine chambers on the bottom of the board, only nine rows. This line defines the number of paths from the center slot at the top of the board to the bottom of the board. Considering that the chips bounce back from the side, it is easy to correct this. Table 1 shows the necessary correction when the board height is 13 rows and the width is 9 rows. The addition to the 12 and 220 added to 66 is a mathematical shown by making a triangle of Pascal after the 13th row of 66 in the 13th row. This is considered to be physically folded on the left side of the triangle, which has a folded line on the vertical line between 12 and 66, on the remaining triangle. Note that 12 line up with 66 and 1 line up with 220. Similar to the right side of the triangle, fold the right side of the triangle with the remaining part of the triangle,	+12	+12
Table 1: Correction of possible paths with a pinker board	In other words, there are 78 ways to enter $ 100 (A 'and I'), how to enter the $ 500 chamber (B 'and J'), and 495 ways to enter the $ 1000 chamber (c (c)). 'And g'), there are 792 ways to enter the $ 10 chamber (d 'and e'), and 924 (e ') are on the $ 5, 000 or $ 10, 000 chamber. The difference in amount varies depending on the year of production. Again, these are the probability of dropping chips from the middle slot. Considering dropping a chip to a different slot at the top means simply moving a broken line.	Research topics 1	The first research task is related to the very interesting concept of mathematics. Even if there is a $ 25, 000 or $ 50, 000 offer, how many producers will be given by whether there is a $ 5, 000 or $ 10, 000 chamber in the middle of the board? The following Table 2 describes the probability required for the calculation. This table shows the probability of falling into each of the nine rooms crossing the bottom when the chip is dropped from the middle groove.	P (a ')	The first research task is related to the very interesting concept of mathematics. Even if there is a $ 25, 000 or $ 50, 000 offer, how many producers will be given by whether there is a $ 5, 000 or $ 10, 000 chamber in the middle of the board? The following Table 2 describes the probability required for the calculation. This table shows the probability of falling into each of the nine rooms crossing the bottom when the chip is dropped from the middle groove.	Research topics 1	In other words, there are 78 ways to enter $ 100 (A 'and I'), how to enter the $ 500 chamber (B 'and J'), and 495 ways to enter the $ 1000 chamber (c (c)). 'And g'), there are 792 ways to enter the $ 10 chamber (d 'and e'), and 924 (e ') are on the $ 5, 000 or $ 10, 000 chamber. The difference in amount varies depending on the year of production. Again, these are the probability of dropping chips from the middle slot. Considering dropping a chip to a different slot at the top means simply moving a broken line.	Table 1: Correction of possible paths with a pinker board

P (f ')

P (g ')

P (H ')

P (I ')

0. 019

0. 054

0. 121

0. 193

0. 226

0. 193

1. 0. 121
2. 0. 054
3. 0. 019

Table 2: Probability of falling into each room

Using a simple expectation calculation as shown below, you can find the expected value that the producer will let go when the chip falls from the middle room.

In other words, the expected value when the player has all five chips is $ 2, 555. 32 per chip, a total of $ 12, 766. 60. This is the expected value when the chip is dropped from the central slot, which is worth $ 10, 000 in the central chamber. This is when the announcer has a chance to get a maximum of $ 50, 000. In actual games, the participants are given one chip, and the probability of acquiring the remaining four chips is 0. 5. In other words, the expected value is $ 2555. 32+(0. 5) ($ 2555. 32)+(0. 5) ($ 2555. 32)+(0. 5) ($ 2555. 32) = $ 7665. 97. Again, this is all about when the chips are dropped from the central room. If the questioner moves to one or the other, the expected value will decrease, and the simulation, part of this study, clearly shows this. In other words, the expected value of the producer is about 15 % of the amount that the announcer may acquire.

1. Research Terms 2 < Span> The first research topic is related to the very interesting math concept of expected value. Even if there is a $ 25, 000 or $ 50, 000 offer, how many producers will be given by whether there is a $ 5, 000 or $ 10, 000 chamber in the middle of the board? The following Table 2 describes the probability required for the calculation. This table shows the probability of falling into each of the nine rooms crossing the bottom when the chip is dropped from the middle groove.
2. P (a ')

P (b ')

P (c ')

P (d ')

P (e ')

P (f ')

P (g ')

P (H ')

P (I ')

0. 019

0. 054

0. 121

0. 193