What are your odds for winning $1,000,000 by playing Plinko during The Price is Right Primetime Special?<\/p>\n
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.<\/p>\n
Here are the rules to win:
\nA. You need to get at least three chips into the middle slot
\nB. If you satisfy A, then you get a “gold chip” which you have to get in the middle slot to win $1,000,000<\/p>\n
What are the factors involved here?
\nA. Odds of the number of Plinko chips you win in the pricing game
\nB. Optimal location to drop the Plinko chip and odds of a Plinko chip falling in the middle slot
\nC. Odds of doing this at least three times to win the Golden Chip<\/p>\n
Well, let’s talk about the assumptions:
\nWe 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.<\/p>\n
A. Pricing Game Odds<\/p>\n
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.<\/p>\n
We can use Pascal’s triangle to help us in determining the odds of the potential outcomes.<\/p>\n
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.<\/p>\n
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.<\/p>\n
| \n Start<\/strong><\/p>\n<\/td>\n \n<\/td>\n \n<\/td>\n \n<\/td>\n \n<\/td>\n 1<\/strong><\/p>\n<\/td>\n \n<\/td>\n \n<\/td>\n \n<\/td>\n \n<\/td>\n<\/tr>\n Item 1<\/strong><\/p>\n<\/td>\n \n<\/td>\n \n<\/td>\n \n<\/td>\n 1<\/strong><\/p>\n<\/td>\n \n<\/td>\n 1<\/strong><\/p>\n<\/td>\n \n<\/td>\n \n<\/td>\n \n<\/td>\n<\/tr>\n Item 2<\/strong><\/p>\n<\/td>\n \n<\/td>\n \n<\/td>\n 1<\/strong><\/p>\n<\/td>\n \n<\/td>\n 2<\/strong><\/p>\n<\/td>\n \n<\/td>\n 1<\/strong><\/p>\n<\/td>\n \n<\/td>\n \n<\/td>\n<\/tr>\n Item 3<\/strong><\/p>\n<\/td>\n \n<\/td>\n 1<\/strong><\/p>\n<\/td>\n \n<\/td>\n 3<\/strong><\/p>\n<\/td>\n \n<\/td>\n 3<\/strong><\/p>\n<\/td>\n \n<\/td>\n 1<\/strong><\/p>\n<\/td>\n \n<\/td>\n<\/tr>\n Item 4<\/strong><\/p>\n<\/td>\n 1<\/strong><\/p>\n<\/td>\n \n<\/td>\n 4<\/strong><\/p>\n<\/td>\n \n<\/td>\n 6<\/strong><\/p>\n<\/td>\n \n<\/td>\n 4<\/strong><\/p>\n<\/td>\n \n<\/td>\n 1<\/strong><\/p>\n<\/td>\n<\/tr>\n Chips<\/strong><\/p>\n<\/td>\n 1<\/strong><\/p>\n<\/td>\n \n<\/td>\n 2<\/strong><\/p>\n<\/td>\n \n<\/td>\n 3<\/strong><\/p>\n<\/td>\n \n<\/td>\n 4<\/strong><\/p>\n<\/td>\n \n<\/td>\n 5<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n And here is a table of the probabilities for each number of chips:<\/p>\n 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.<\/p>\n B. Optimal Location to Drop the Plinko Chip<\/p>\n 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?<\/p>\n 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<\/a>). 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.<\/p>\n Here are the expected slots that the chip will fall in out of 4096 possible paths, when dropped off in the middle.<\/p>\n So, we finally have our magic formula: it has to be dropped from the middle slot<\/strong>, and it’ll end up in the middle slot 924 out of 4096 times, or about 22.6%<\/strong>.<\/p>\n C. Landing in the Middle at Least Three Times<\/p>\n 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<\/a>) from Professor Richard Lowry of Vassar College to learn about this subject.<\/p>\n Here’s the formula:<\/p>\n Where: Let’s start with one chip and work our way up.<\/p>\n One Chip – 0% Two Chips – 0% Three Chips – 1.1522% Four Chips – 3.8206% Five Chips – 7.9526% 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.<\/p>\n Add it all up, and your overall odds for getting at least three Plinko chips in the middle is a whopping 1.8843%<\/strong>!<\/p>\n 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.<\/p>\n 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%<\/strong> In other words, if you had 235 playing Plinko, only one would win the million.<\/p>\n If we were to assume that the players were all-star pricers and won five chips every time, the odds would be 1.7940%<\/strong>, or one out of every 55 people who played.<\/p>\n 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%<\/strong>.<\/p>\n 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.<\/p>\n How was my math? Did I do it correctly? Comments and feedback appreciated.<\/p>\n Props go to Mr. Weicker for helping me figure out some of the binomial probability stuff.<\/p>\n Resources: 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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[50],"tags":[75,74],"class_list":["post-135","post","type-post","status-publish","format-standard","hentry","category-culture","tag-games","tag-plinko"],"_links":{"self":[{"href":"https:\/\/www.coreyhulse.com\/corey\/wp-json\/wp\/v2\/posts\/135","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.coreyhulse.com\/corey\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.coreyhulse.com\/corey\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.coreyhulse.com\/corey\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.coreyhulse.com\/corey\/wp-json\/wp\/v2\/comments?post=135"}],"version-history":[{"count":4,"href":"https:\/\/www.coreyhulse.com\/corey\/wp-json\/wp\/v2\/posts\/135\/revisions"}],"predecessor-version":[{"id":1005,"href":"https:\/\/www.coreyhulse.com\/corey\/wp-json\/wp\/v2\/posts\/135\/revisions\/1005"}],"wp:attachment":[{"href":"https:\/\/www.coreyhulse.com\/corey\/wp-json\/wp\/v2\/media?parent=135"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.coreyhulse.com\/corey\/wp-json\/wp\/v2\/categories?post=135"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.coreyhulse.com\/corey\/wp-json\/wp\/v2\/tags?post=135"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}} |