your odds of winning $1,000,000 by playing plinko

July 25th, 2008 · 7 Comments

What are your odds for winning $1,000,000 by playing Plinko during The Price is Right Primetime Special?

I thought about this the other day as I was watching The Price Is Right at night, where they have two chances on every show to win $1,000,000. On this particular show, one of those opportunities came during Plinko.

Here are the rules to win:
A. You need to get at least three chips into the middle slot
B. If you satisfy A, then you get a “gold chip” which you have to get in the middle slot to win $1,000,000

What are the factors involved here?
A. Odds of the number of Plinko chips you win in the pricing game
B. Optimal location to drop the Plinko chip and odds of a Plinko chip falling in the middle slot
C. Odds of doing this at least three times to win the Golden Chip

Well, let’s talk about the assumptions:
We are assuming that when a chip hits a peg, there is a 50% chance that it’ll go either left or right. This does not take into account potential things like faulty pegs which have a favoritism to one side. In also does not take into account inertia, which is to say that if a chip were to come in from a particular direction with momentum that the odds of it continuing in that direction when it faces the choice of left or right would be biased. We are going to keep the assumption that the odds are 50%. If anyone who has a physics background wants to contribute to the conversation with an analysis of the inertia of chips on a Plinko board, by all means send me an e-mail and together we can revel in our nerdiness.

A. Pricing Game Odds

In order to win Plinko chips, you are given one chip to start and are presented with four items, each with a price, and you have to guess whether or not you think the tens digit or the ones digit is correct in the price. If you were randomly guessing, you’d be right 50% of the time. For each one you get right, you get a chip. Now, in order to even have a chance at winning $1,000,000, you need to have at least three chips.

We can use Pascal’s triangle to help us in determining the odds of the potential outcomes.

You are at the first item, and you have a 50% chance of getting it right. Looking at the chart below, if you get it right, you move right on the path, if you get it wrong you move left.

Whether you are right or wrong, you move on to a chance at the next chip (and down another level on the triangle). This goes on until you’ve reached the fourth item, in which you now have five possible outcomes and sixteen different paths to get to those five, each with a possible percentage based on the odds.

Start

1

Item 1

1

1

Item 2

1

2

1

Item 3

1

3

3

1

Item 4

1

4

6

4

1

Chips

1

2

3

4

5

And here is a table of the probabilities for each number of chips:

Free Chip Won Chips Percentage
1 0 1/16
1 1 4/16
1 2 6/16
1 3 4/16
1 4 1/16

So, 11 out of 16 times, you will have enough chips to potentially get a chance at the gold chip. However, looking ahead, there are going to be different odds for winning the gold chip depending on how many starting chips you had.

B. Optimal Location to Drop the Plinko Chip

Obviously, in order to maximize your odds, you have to think about the optimal place to drop it. For anyone who understands probability distribution, then this it’s pretty clear that you want to drop it directly over the slot that you’re attempting to land it in. However, one of the questions that I had involved the reflecting of a chip off of a wall. Do these reflected amounts actually change my odds to favor a different place to put it?

It turns out the answer is no. There is a very interesting site that I stumbled upon which has done all of the math already on why this is so, so credit goes to Susie Lanier and Sharon Barrs from the Mathematics and Computer Science Department at Georgia Southern University (http://mathdemos.gcsu.edu/mathdemos/plinko/bigboardplinko.html). The assumption that they make is that the board is a 9 x 13 configuration of pegs, so when looking at the Pascal’s triangle listing of the possible paths a chip could take, playing the chip right over the middle slot will yield the best results. In fact, when you play your chip directly in the middle, only 26 out of 4096 possible paths will bump off of the wall. Also on the site are instructions on how to build your own Plinko board, along with some JavaScript simulations of the game.

Here are the expected slots that the chip will fall in out of 4096 possible paths, when dropped off in the middle.

Slot Paths Odds
1 66 1.6%
2 220 5.4%
3 495 12.1%
4 792 19.3%
5 924 22.6%
6 792 19.3%
7 495 12.1%
8 220 5.4%
9 66 1.6%

So, we finally have our magic formula: it has to be dropped from the middle slot, and it’ll end up in the middle slot 924 out of 4096 times, or about 22.6%.

C. Landing in the Middle at Least Three Times

Now that we know how often it’ll land in the middle, we have to figure out the odds of doing it at least three times. For our final formula, we need to figure out this number for each possibility of the starting number of chips we have. For this, we will use binomial probability. Here’s a reference guide (http://faculty.vassar.edu/lowry/ch5apx.html) from Professor Richard Lowry of Vassar College to learn about this subject.

Here’s the formula:

P(k out of N) = N!


k!(N-k)!
(pk)(qN-k)

Where:
N = the number of Plinko chips
k = the number of times we hit the center
p = the probability that we will hit the center
q = the probability that we will hit any place but the center

Let’s start with one chip and work our way up.

One Chip – 0%
Well, we’ve only got one chip, and we need to hit the center three times. Looks like an automatic fail to me.

Two Chips – 0%
Same deal here as before.

Three Chips – 1.1522%
We have the right requirements for this calculation.
N = 3
k = 3
p = 924/4096
q = 3172/4096
Plug and chug, and you get your answer.

Four Chips – 3.8206%
N = 4
k = 3 and 4
p = 924/4096
q = 3172/4096
Here, you have to do two calculations and add them, one to account for getting three out of four (3.5560%) AND for getting four out of four (0.2590%) in the middle.

Five Chips – 7.9526%
N = 5
k = 3 and 4 and 5
p = 924/4096
q = 3172/4096
Three out of five (6.8844%) + four out of five (1.0027%) + five out of five (0.0584%)

So, to bring this all together, let’s bring back out other chart where we list out the odds of obtaining the chips and combine it with the probabilities of hitting the center the required number of times to win the golden chip.

Free Chip Won Chips Chip % Middle % Calculated Odds
1 0 1/16 0% 0%
1 1 4/16 0% 0%
1 2 6/16 1.1522% 0.4321%
1 3 4/16 3.8206% 0.9552%
1 4 1/16 7.9526% 0.4970%

Add it all up, and your overall odds for getting at least three Plinko chips in the middle is a whopping 1.8843%!

So, we’ve figured out the odds of earning a golden chip. Now what we have to do is land that golden chip in the middle slot and we’ve won the million.

The formula for that is simple enough now that we’ve done all the calculations. We need to multiply our odds of getting here in the first place (1.8843%) by the odds of the chip falling in the middle once (924/4096), and you have your odds of winning $1,000,000 by playing Plinko: 0.4251% In other words, if you had 235 playing Plinko, only one would win the million.

If we were to assume that the players were all-star pricers and won five chips every time, the odds would be 1.7940%, or one out of every 55 people who played.

The only real “factor” here that you can control is how well you play the pricing game. If you’re not a random guesser and you at least know something about the prizes being offered, then you odds increase in terms of winning chips to begin with, so the range that you would expect a decent pricing game player to fall in is anywhere between the two numbers we calculated, 0.4251% to 1.7940%.

And despite all this, Plinko is still my favorite game. I am (hopefully) going to see a taping of The Price Is Right next month, and if I get the chance to be on stage, I absolutely want to play Plinko.

How was my math? Did I do it correctly? Comments and feedback appreciated.

Props go to Mr. Weicker for helping me figure out some of the binomial probability stuff.

Resources:
http://mathdemos.gcsu.edu/mathdemos/plinko/bigboardplinko.html
http://faculty.vassar.edu/lowry/ch5apx.html

→ 7 CommentsCategories: culture | Tags: , | 9,776 views

7 responses so far ↓

  • 1 jerricka // Jul 26, 2008 at 5:57 pm

    This is just ridiculous. How long did it take to calculate?

  • 2 Ben // Aug 6, 2008 at 9:01 am

    As someone who’s not particularly impressed by Plinko and has only watched about four games played, I’m surprised you missed a very common feature – in at least two of the vids I watched, at one point in the drop the disc scooted left or right by two pins in one row. I’m not sure how this affects the maths since I suppose it could happen in either direction, but it certainly complicates things.
    Also, it may be the limited sample that I’m seeing, but for most of the drop the disc seems to take a regluar left-right-left-right pattern, possibly due to being slightly smaller than the holes? That would certainly make the centre more favourable, but I suppose it’s well into the realms of asking a physicist since you have to consider how much of a fall is needed to reach a speed where it can jump out of that rhythm.

  • 3 blog and 365 updates — CoreyHulse.com // Feb 14, 2009 at 12:11 am

    […] my “Best of” based on the posts that seem to still get a lot of hits (like my plinko post) and then other posts that focus on my various […]

  • 4 Quixzotek // May 26, 2011 at 10:00 am

    I have a question for you. I was at a casino the other day and a lady had won a chance to win money at Plinko. I was helping her decide where to drop from, moving her left and right to what I felt was the highest probability area. There were 9 chips given to her in a 9 slot board. The middle slot was $2,500, the two on either side was $1,000, then out from there was $500, then out from there $300, then the two corners were $200. So, the idea in this particular game was you won the money the chip dropped in until you hit the same slot twice. The total would be added and you would win that. If somehow you put a chip in every slot perfect you would win $25,000. Well, she watched me guide her the whole way, moving her inches left or right. She would place the chip above the board and I’d guide her until she had the position I wanted.

    Well, the plan was going well until finally she had one chip left and all slots were perfect. I looked at her, she was about to drop and I moved her about 2 inches to the left. She was already guaranteed $6,000, but this final dropped chip could win her $25,000. Well, she dropped it where I told her and it clinked down left, then right and guess what, fell right into the final slot and she won $25,000. The bad news is she only gave me $100 for my instructions. She had no clue what she was doing and I felt like $500-$1,250 was more appropriate. But, nevertheless, my question is this as I can’t find the answer.

    With a 9 slot Plinko game with 9 chips to drop, what are the odds of filling all the slots perfectly, one into each? Please send the answer to my email as I’m very curious to know. The casino manager had told me she was the first one to ever accomplish that feat in the 6 years of their existence!

  • 5 Damaris // Jul 23, 2013 at 11:01 am

    Let’s do it. s bassist on ”. Depending about what you sell, preparing your goods will take a number of hours weekly or perhaps a full-time job.

  • 6 BillQ // Sep 27, 2013 at 8:04 pm

    Your link to Georgia Southern University is gone

  • 7 BillQ // Sep 27, 2013 at 8:04 pm

    Your link to Georgia Southern University is broken

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