One year ago I sat at a sports bar in London as the Mets lost their final game of the season to the Florida Marlins which capped their historic collapse in which they blew their seven-game NL East lead with 17 games to go.

Yesterday, the final day of the regular season, the Mets found themselves in a similar situation.  They needed to win the final game of the year to put them in the position of either forcing a one-game playoff or making the post-season outright.  Well, long story short, the Mets lost to the Marlins, and are the first team to ever miss the playoffs two years in a row by losing their last game at home on the final day of the season.  This proves that while the Mets are a pretty good team for 95% of the year, they do not have the ability to command the loyalty of sea creatures.

Last year everyone on the official Mets e-mail list got an apology from the organization for the collapse.  This year, not only do I expect an apology e-mail, but I also expect free tickets to the 2013 Home Run Derby in which the Mets are rumored to be hosting at their new stadium.  Make it happen, Mets.

Zoidberg the Betta Fish, a life-long resident of Philadelphia, passed away from natural causes this week.  He was 1 3/4 years old.

Named after the fictional doctor in the “Futurama” series, he will be remembered for his weekly-changing monikers, which included “Foreign Correspondent Zoidberg”, “Assistant to the Regional Manager Zoidberg”, “Hot Rod Zoidberg”, and “Supreme Mugwump Zoidberg”.

Zoidberg’s favorite meal were Betta Pellets.  His favorite hangout was the green plastic plant, where he would often be found resting on the leaves.  He was also known to frequent to the orange plastic plant.

In lieu of flowers, the estate of Zoidberg wishes for you to share your memories of him on this website.

Here’s the August version of “Songs That Are Stuck In My Head When I Wake Up”.

Strangest song this month? “Here Comes The Hotstepper” wins by a landslide.

And since I’ve been at this for four month, I figure an Excel graph mapping out the genres of the songs was appropriate. That, I and I really like Excel.

When I was in London, I picked up an “Aurora Color Changing Alarm Clock”.  It changes colors every hour, so I decided to take a picture every hour to put together a collage of the colors because I thought it would look cool, and it would help remember what colors would go to each hour.

However, I’m disappointed to find that 7 and 8 are close in color, as are 9 and 10, and 11 and 12.  It’s like they started programming the clock with a variety of different colors for 1 through 6, and after that they got lazy and said, “eh, forget it, let’s make the rest of ’em all a shade of blue”.

Below is the collage.

Last week was my first time in California.

I’ll (hopefully) be writing up more later about the trip, but we ended up at Balboa Island.  If my calculations are correct, we ended up about here.  The water wasn’t terribly cold on that day, so of course I had to run in.  I didn’t travel over 3,000 miles to just stand and let the water hit my feet.

Of course it wasn’t until after I got out of the ocean that I realized that we didn’t pack any towels…

For those who don’t know, woot is a website that pioneered the “Deal a Day” concept, where starting at midnight Central, a new item goes up for sale until they either sell out or it hits midnight again and the pumpkin will turn into something new, like a talking animatronic Elvis.

Last year they jumped into the world of t-shirts, and starting offering shirt.woot, a “T-shirt a Day” website.

Today’s Play of the Day goes to the t-shirt featured on Sunday, which has a fictional brand of soda called “F5”, whose tag line is “It refreshes you!”

I of course bought one, and for a limited time you can too.  On the first day it’s offered, you can nab it for $10, but after that, you can buy it for a limited time for $15 until the shirt is no longer popular, at which point they stop producing them.

This marks my second shirt.woot purchase, the first one being one of the very first shirts from when the site launched a year ago.

So this past week I had the good fortune of being back in London, and I’m currently typing this as I sit on my plane back to the States. (Note: It took me about a week and a half after I got back to actually post this.)

While a week is hardly sufficient time to have adventures like those of my two-month stay here, I still managed to get around town, eat some really good food, and explore some parts of London that I hadn’t been to before.

One thing to note before I go on.  For those of you who lived in Stouffer College House at the University of Pennsylvania, you no doubt have come across the maroon Stouffer pens during your stay here.  Let me tell you, they do NOT travel well in high altitudes.  I had multiple pens explode on me while airborne.

Trip details and some pictures after the jump. [Read more →]

So here’s the next edition of songs that are stuck in my head when I wake up.

July 26th had two songs because I woke up, had “Down on the Corner” in my head, then fell back asleep and woke up with “Renegade” on my brain.  Strangest song of the month: Waterloo by ABBA.

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.

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

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.

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:

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.

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

So, a few months ago I joined Twitter.  I haven’t used it yet, but today I decided to spruce up the website by adding a “Twitter” section, where it imports the latest post from my Twitter RSS feed.  Currently, only two of my friends that I know are on Twitter, so if you’re on Twitter too, come find me.

http://twitter.com/coreyhulse