Math post: Probability

Here is a long needed mathematics blog post. This weeks subject is Probability and the question is as follows:

Suppose that a biased coin that lands on heads with probability p is flipped 10 times. Given that a total number of 6 heads result, find the conditional probability that the first 3 outcomes are

(a) h,t,t (meaning that the first flip is a head, the second is a tail, and the third is tails)

(b) t,h,t

Answers:

(a) We will have to know one formula to calculate these probabilities, as well as some basic knowledge on summations and the choose function. The needed formula is P(A|B)= P(AB)/P(B). This reads as the probability of an event A given B is equal to the probability of A and B both occurring over the probability of B occuring.

Thus for (a), we are trying to find P(H,T,T|6 Heads). We have:

P(H,T,T|6 Heads)= P(H,T,T and 6 heads)/ P(6 heads)

=P(H,T,T)*P(6 heads |H,T,T)/ P(6 heads)

since P(6 heads|H,T,T)= P( H,T,T and 6 heads)/ P(H,T,T) implies that P( H,T,T and 6 heads)=P(6 heads|H,T,T)* P(H,T,T)

= (p*(1-p)*(1-p))* (7 choose 5)*p^5*(1-p)^2) / (10 choose 6)*p^6*(1-p)^4

The logic in this is first give the probability of H,T,T. That is give the probability of flipping heads (p) times the probability of flipping tails twice (1-p)(1-p). (from now on I will write q for 1-p) Thus we have P(H,T,T). Then we must solve for P(6 heads |H,T,T). Think about how many ways the other five heads can be given if we have already been give one. We have already done three of the ten flips, so we have seven more and we need to get exactly five heads in those flips. Thus we get the (7 choose 5). We then need the probability of getting exactly 5 heads and 2 tails: p^5*p^2. Thus we have the probability of P(6 heads |H,T,T). For the probability of 6 heads we use the same logic as before but now no flips have been used, so we have ten flips to land 6 heads. There are (10 choose 6) ways of doing this. We then consider the probability of this occurring. This is p^6*q^4.

We thus receive the equation:

=(p(q)(q)(7 choose 5)(p^5)(q^2))/ ((10 choose 6)(p^6)(q^4))

=((p^6)(q^4)(7 choose 5))/ ((p^6)(q^4)(10 choose 6)) by combining the p's and q's (remember (x^a)(x^b)=(x^(a+b)))

=(7 choose 5)/ (10 choose 6)

=21/210 (calculator or (n choose i)= (n!/ (n-i)!(i!)) where n!= n(n-1)(n-2)...*2*1 ¹)

=1/10

(b) P(T,H,T| 6 heads)= P(T,H,T and 6 heads)/ P(6 heads)

=P(T,H,T)*P(6 heads| T,H,T)/ (P(6 heads))

=((q^2)(p)(7 choose 5)(p^5)(q^2))/ ((10 choose 7)(p^6)(q^4))

=((p^6)(q^4)(7 choose 5))/ ((p^6)(q^4)(10 choose 6))

=(7 choose 5)/ (10 choose 6)

=21/210

=1/10

¹ (7 choose 5) = 7!/ (2!)(5!)= (7*6*5*4*3*2*1)/ (2*1)( 5*4*3*2*1)= 7*6/ 2= 42/2=21

(10 choose 6)= 10!/ (4!)(6!)= (10*9*8*7)/ (4*3*2*1)= 5040/ 24=210

I hope that you enjoyed this. The next post will be a cultural one. I hope to be posting math based posts at a higher rate in the future, so for those who are interested "Yay!", for others... sorry, but I hope it makes you feel better that I am able to understand the material better if I work to break it down for you, so please let me know if you need any thing else explained in more detail. 

Barcelona, Spain!

 I have just returned from an amazing weekend in Barcelona, Spain. The weekend consisted of many things from visiting sites like Sagrada Familia, Parc Guell (both designed by Gaudi), the old city (El Gothic), Fira de Barcelona-Montjuic, Torre Agbar, the old Olympic stadium, eating tapas, getting to know many great people in our hostel, visiting the beach (and the clubs), and much more. It was a very busy weekend, and although the weather did not seem to agree with us, we all (Sheri my roomamte, Sara our friend, and myself) had an amazing time.

Us at El Gato Negro

The first day in Barcelona was pretty hectic seeing as we arrived on Friday after noon. We all had attended classes Friday morning and then rushed to the airport to catch our flight. The flight was quick although fairly stressful. I usually am pretty comfortable with flying, but something about this flight made me very nervous. I think it may have been that I was flying with Ryanair, a flying service that I had never really heard much about it terms of reliability. Before take off I read every emergency procedures product I could get my hands on and managed to tear my nails apart pretty badly. A bit of a bumpy take off made me really regret giving up my window seat (I prefer to watch the takeoff than just feel it). Once we had settled down, I read through a magazine that the flight attendant had given me and had some good conversation with a group of people that were from Barcelona. I asked them how they liked Budapest, and they gave me a long list of things to do while in Barcelona. 

My marshmallow drink at El Gato Negro

Upon arriving, we unpacked, grabbed some food at a local buffet and headed out to a bar called El Gato Negro (The Black Cat). There the environment was great. Many of the drinks consisted of fire tricks. For example, one that a nice guy bought for me (who also happened to be gluten intolerant) consisted of spraying the counter with alcohol, lighting it, us toasting marshmallows over the fire and then dipping them into our very sweet drinks and eating them along with the rest of the drink. It was unbelievably good. My friends and I then bought a litter of Sex on the Beach to share as we hung out with the guys some more. They were very sweet, but we left shortly after they did to head to bed to get sleep for the long day ahead.

What the finished Sagrada Familia
 should look like

The next day was much more exciting. It consisted of visiting Sagrada Familia and Park Guell. Sagrada Familia is a very large church that has been under construction for over 100 years now and it still has much more time until it will be finished. It consisted of four large towers (with the fifth largest under construction), many arches, images of people, war scenes, Christmas trees, and much more. It was not opened on the day that we visited to enter, but we did spend much time inside the shop. I bought multiple post cards (I have been collected them from everywhere I go) and a small Gaudi designed turtle. 

Torre AGBAR

We also visiting Torre AGBAR, which can only really be described as a large phallic object. We did not know at that point that the tower lit up and that we should have visited at night. We then went to Park Guell (and got lost along the way) which is a large park outside that was also designed by Gaudi. It took us over two hours to reach Park Guell because it is located in the outskirts of the city and we managed to get lost multiple times. Gaudi had lived in Park Guell when he was in his twenties. The park consisted of many outdoor paths and natural looking constructions. There were also many mosaic and metal pieces placed throughout the park. There was an amazing panoramic view of the city from the top of Park Guell. After visiting Park Guell, my friends and I went back to the hostel where we relaxed for a little while (I took a nap) and then we headed out once again to meet some other friend from BSM. We all met up at this great meat restaurant where you could order many different types of meats and cheeses for fairly inexpensive prices. I got a plate of sausage, some cheese, and a very good glass of wine (wine is starting to grow on me here) for only 7 euros (about 11 USD). We then walked with them through a little shop area and sat with our vegetarian friend while she ate a pizza from a nearby restaurant. We then split as my group went to go see the clubs by the ocean and the other group went to check out some more bars. 

Me at Park Guell

One of Gaudi's most recognized pieces

Upon arriving at the club area we realized that we were there far too early so we went and walked along the ocean and talked some. We then went to search for any clubs that may be starting early. There were none. The area was completely deserted. Even though we knew that if we stayed longer that we'd be lost without a way home besides taking a cab, we decided to stay and wait things out. An hour passed before anything started to start up a bit, but even then all of the parties were private preparties that we were not allowed to enter. So, we sat out in the rain for another hour or so until things started to get going more (it was about 12:30 at this point). We bounced between some bars but no one was dancing and most were completely empty. Finally we found a place to dance, but after a while we realized that the music was not very good and that most of the people there were left over from the preparties. We all felt a bit out of place (or at least I did) and decided to leave. After trying desperately to find any other open bars, we finally decided to cut our loses and head home early. It was rather disappointing to not be able to experience that Barcelona night life, but honestly it probably would have cost too much and required us to stay out far too late for it to have been worth it.

The next morning we all slept in a little and then headed out to Fira de Barcelona-Montjuic. This was an area with many interesting pieces of architecture. There were two towers that looked like they may have at one time formed a gate, multiple fountains, a row of four white columns, and finally the castle which has now been converted into an art museum. We walked around for a long time taking pictures and admiring the wonderful view from the top of the castle. The weather was perfect. We then decided to continue up the stairs where we found a botanical garden and the old buildings from the 1992 Olympics. There was a small festival going on for little kids, so we walked around some and then headed down and back home. We rested a little and then headed back out for a walking tour of the old city. It was free, and surprisingly it ended up only being my friends and I who attended. The tour guide was very friendly and knowledgeable. We visited many churches and learned much about the Catalonia region and the history of Barcelona in their fight for independence.

Catalonia is a region older than Spain that has fought for its independence from many different rulers (Muslims, French, Roman, Spanish, and many others) for many years. They have not, however, really ever been successful. Currently Catalonia is fighting for its independence from Spain because it is one of the richer areas and because Spain is currently in a very tough economic time much of Catalonia is supporting the rest of the city. This area even speaks its own language, Catalan, and has its own flag. The history of Catalonia is very interesting and I highly recommend looking into it.

For dinner we all went to get tapas (a requirement while visiting Spain). My friends both got the six tapas sampler and I got three different tapas- patatas bravas, a chorizo and potato dish, and a chickpea and baby shrimp fritter. All of it was unbelievably delicious, but I must admit that the shrimp fritter was my favorite (a surprise seeing as I am not normally a shrimp fan- another thing that must be growing on me). The waitress was very sweet and I had multiple conversations with her in Spanish. After finishing we had intended to go see two light shows, one that the tourguide had suggested and then the Torre AGBAR lit up, but that turned out to not be what the night had intended for us.

Then a very interesting story began. We all headed towards the metro, but then we entered the wrong side of the station and had to switch over. Because we had already swiped our cards, the machines would not let us enter so I called a security guard over to help us. I was talking to him and upon turning around I noticed a man trying standing behind my friend Sara and her bag was open. Within a few seconds I recognized what he had been trying to do and well... I went into defensive mode. I said a loud “Uh!” and stepped towards him forcefully with a very sharp look on my face. Sara's face went white in fear as she did not realize what was happening. The man immediately froze and held out his book bag to me saying in Spanish that he didn't take anything. The security guard rushed over and began to check his bag asking my friend in Spanish “Tienes todo?” (Do you have everything?) which she did not understand. She did not realize what was happening and for what felt like a minute she fumbled with her bag saying “Yeah, I think so” as I prompted her to keep checking. Finally she agreed that she had everything as the metro pulled up. I told the guard that she had everything and to please let us through. He did and we rushed to catch the metro. It was very stressful and I scolded Sara because Sheri and I had been telling her all weekend to carry her purse in front of her yet she kept failing to do so. It really does amaze me how unaware some people can be of their environments. She really got lucky to have not lost anything, and I'm glad that I know how to handle myself in that sort of situation (thanks dad <3 ).

Church during the tour of the Old City

We left the metro and within a few minutes it began to pour rain. We ran to the nearest covering and stayed there to decide what we were going to do. The light shows were now out of the picture, so we decided to rush back to the metro and head back to our hostel. We rushed through the rain and, luckily, did not have any other incidents while entering the metro. We arrived home and the rain had stopped. We waited for a short time to dry off and gather our umbrellas and then we headed out to find Sara a crepe and Sheri and I maybe some ice cream. I found this little shop that had both and actually also had gluten free cones. I ordered my first ice cream cone in probably three years now with two scoops- dulce de leche and a cinnamon meringue. It was absolutely delicious and was an amazing way to end the trip.

Another church during our tour

Even though everything did not work out perfectly for this trip, I still have a great time in Barcelona. Visiting the city really opened up my mind to maybe living in Europe in the future. I love that I got to practice my Spanish because, for me, speaking Spanish is one of the greatest thrills. I got so many compliments and had some amazing conversations. Every day I get better and better. I feel like a bit of a different person when I speak Spanish. I feel much more mature and happier. I would love to return to Barcelona some day, maybe when its a bit warmer so that we can visit the beach. If the economy picks up there maybe I could even consider it for more than just a visit but as a future home.

I hope that you enjoyed this blog on my trip to Spain. There will be another math post tomorrow. Pictures will be posted on my Facebook account that you should all be able to see. If we are not friends, then we should change that. <3

Week of hell

So this past week for me has been a roller coaster. I began the week with being unbelievably stressed because I has spent the previous Friday and Saturday in Szeged (the third longest city in Hungary) at a lecture and visiting the city, and then I spent Saturday night at the Sausage and Palinka festival at Buda castle. It was an amazing two days, but because I was so busy, I had little time to rest nor to work on the multiple homework assignments I had nor to study for my two exams. Because of this, I ended up feeling very behind and stressed on Sunday and Monday. I also had been having a very hard time adjusting to the amazing levels of intelligence of the group of students at this program. So many of them are in far more classes than me and many seem to enjoy constantly thinking of math. They remind me of how I used to be.

When I was in high school, I adored math. It was probably my favorite thing in the world, with Mexican culture being its only competitor. I was in math team and absolutely loved it. Practice was always the highlight of my day. But ever since coming to college and taking multivariate calculus (which I hated), my love for math has continued to decline. I have found myself thinking in a much more qualitative manner, and ideas that previously would have been simple for me to grasp have become extremely difficult. I have had to fight for every grade, and the main reason I have continued math, and why I decided to attend this program, was the hope that maybe I would reacquire my love for math. In many ways, I have. I love the challenge that is math. I love the different way of thinking that I know many people cannot grasp. I love the fun little tricks to solving extremely complex problems. I love all of that... but I have found that upper level math is nothing like calculus and the “math” that most people think of. Proof writing, such abstract ideas, and complex theorems that as common in higher level math have never caused as much excitement in me as concrete math with actual numbers. Even considering this, I have really enjoyed my time here studying set theory, graph theory, probability, and MPS. Even with the extremely frustrating problems and long nights, I have begun to love math again. This week, however, has made me question that love again.

Because I was so far behind at the beginning of this week, the rest of the week as been a crazy race to catch up. Tuesday I was up late working on graph theory (for Thursday) and set theory (for Wednesday). Because of this, I had little time to study for my MPS test (Wednesday) that I had assumed would be a breeze. You know the saying that assuming with make and ass out of you and me? Well, I don't know about you, but it made a huge ass out of me. I did much worse on the MPS exam that I had expected. Come test time, many of the processes that I had learned flew out of my mind, and I resorted to very long and drawn out attempts to answer any question I could. I am very lucky that point received on homework goes towards the exams as “extra credit”, because without that I would probably have failed this exam. Even with it, I will be lucky to get a B. Wednesday, after class, I was again up late working on my graph theory homework. That left me only Thursday to finish my probability problem set and to study for my probability exam. I was up late, but come time for the exam, my problem set was complete and I was feeling confident. I had memorized all of the theorems, done all of the sample problem tests from the book, reread my notes, and looked over my previous homework. I thought that nothing would surprise me. Again, I was wrong. The test was based on two problems. One was very similar to many problems we have done before, and I did fairly well on that one, but the other was a very complex expansion on a type of problem that we had only done once before and in a much simpler form. This problem asked for the number of ways that one could get from the origin (0,0,0) to the point (10,11,12) while one is only allowed to move in a positive direction (right, forward, or up). The other problem we had done was “how many ways can you get from (0,0) to (4,3) if you can only move right or up?”. In this problem, I had simply written out all of the possibilities and counted them up. It was a simple grid so I could do that. In the problem on the test however, there was no way for me to do that. There were just too many possibilities- somewhere in the ten thousands. The problem had 5 parts and all of them required the answer from the first which I could not figure out for the life of me. A combination of frustration and a lack of work space made me panic, and I have no confidence in my test results. The only comfort I have is that many other people seemed to be struggling as well, so if I'm lucky, maybe the professor with curve the test. I can only wait and see.

A combination of the frustration with my work, my test results, the realization that I really am no longer the math nerd that I used to be (and was proud of), an extreme missing of my friends and family from back home, and multiple fights with multiple people have made this week a living hell. I have no clue how I did on either of my exams, and I can only hope that I passed both. If I didn't, then I may need to drop or audit a class which will horribly affect my ability to graduate early. At this point though, everything is behind me and there is nothing that I can do about it. I am realizing (with the help of my amazing boyfriend Dante) that not graduating early really won't be the end of the world and that I put far too much pressure on myself (and others) with regards to school. At this point, I can only accept that this has been a very hard week, but that I have made it through it. Hopefully, the ones ahead will be easier. If not, then I'm sure that since I made it through this one that I can make it through the others. Or I can hope...

I'm sorry for such a depressing post this week, but this is suppose to give you a taste of my life abroad, and I feel that not sharing the negatives with you would be an insult to those of you who honestly want to know how I'm doing. At this point, I'm not doing too well. I am however, still very much enjoying my classes and my time abroad. I have learned about myself by being here, and I only expect to learn more. I will be okay, and I am not in any way giving up. I will not drop any of my classes until the very end and I will fight to pass all of them.  

Life on the Danube

Classes are going really well. I'm really starting to love the subject of mathematics a lot more. The professors here are all great and very enthusiastic. I definitely don't think that this is going to stop me from pursuing anthroplogy for grad school, but it does open up my eyes to how I can use mathematics in the field of anthropology, and also that I may want to do some work with mathematics outside of anthropology in the future. I'm considering maybe also working as a tutor or a math team coach at a local highschool. I feel like there is a lot that I can do with my knowledge in mathematics as a second supporting job. 

I am really starting to fall in love with the city of Budapest. I have always felt as strong connection to rivers, and I love that the Danube runs right through Budapest (dividing the city into two cities, Buda and Pest). I love how close everything is, the great public transport, the wide variety of things to do (operas, bars, hookah bars, clubs, public baths, festivals, sites to see, places to work out/ run, zoos, etc) and all of it is fairly inexpensive! I don't know how realistic it is, but at some point in the future I would love to return to Budapest and possibly study the culture here more deeply. I will be keeping my eyes open for anthropological research projects that may involve Budapest in the future. 

One of my main tasks in visiting Hungary was to try and visit the town of Molidorf. This town no longer exists, but its ruins hold the remains of many of my ancestors. Molidorf was actually a German concentration camp in the years after WWII. I have read a lot about it from the site Molidorf.com. I think that it is important for me to visit Molidorf and honor my ancestors there, but sadly I do not think that this is possible. I have done much research on Molidorf and found that it is located in a very very small town on Serbia. The only way that I could make it down there would be to take a train to the border and then find someone to drive me from the border to Molidorf. This, however, seems very difficult. The price of having someone driving me the two hours there, stay with me during my visit for a couple hours, and then drive me back would cost me hundreds of dollars. I have also found that the ruins of Molidorf are located on an abandoned field, and the roads to access this field are hard and at times nonexistent. I would need a tractor or a large truck to be able to access the ruins. Entering Serbia would be very dangerous to me seeing as the US was at war with Serbia just ten years ago and there are still very negative feelings there towards Americans. Also, the field that I would be entering in wooded and used for hunting during this time of year. Because of that, I would be putting myself at high risk by both entering Serbia and by entering the field to visit the ruins. I have not completely rulled out all ways for me to enter Serbia, but at this point it's looking like it is very unlikely that I will be able to make it. With winter coming soon, my only real chance to make it during this trip is within the next few weeks. After that, the roads will become even more treacherous. I will be keeping my eyes open for any opportunity, but visiting Molidorf may have to be delayed to a later point in my life when maybe the situation has changed such that there are more ways to enter Molidorf during a different season of the year when hunting will not be such a large issue. 

I have really been enjoying my time here in Budapest. It has reminded me in many was that I am Hungarian and I should be proud of that. I have fallen in love with this city, and with this city I have begun to be more confident in myself and my heritage. Even though I have greatly enjoyed my time thus far, I must admit that I do really miss a lot of people from back home. I miss my boyfriend Dante so much that it's at times hard to bare. I miss my family and my friends. I miss being at Lawrence (my university). I miss american food and being able to cook more of the meals that I love so much. I miss my friends. I'm sad that I won't be able to see one of my closest friends Brian Kopas before he leaves to join the military on December 3rd. I'm sad that I'm missing the welcoming of all of the new students to Lawrence University- international and domestic. I'm missing so many events, like Michelle Obama coming and speaking at my university, many dances, and performances. I'm not an RLA (Resident Life Advisor) now because I'm here studying abroad. At times all of the things that I'm missing can really pull me down... but honestly I wouldn't change a thing. I love Budapest and I love this opportunity for me to be here and experience a different culture. I love my classes and my new friends here. I also remember that everything I'm missing now will be there when I return. My friends and family, Dante, the new students, more dances and performances, possibly being chosen to be an RLA again, etc. All of these opportunities will still be there when I return to the states and to Lawrence. Because of that, I'm going to work hard to not let these things pull me down and I'm going to make the best of my time here in Hungary. I know that things are just going to get harder, but that I am strong enough to make it through them. I'm really glad that I have you all reading this, and that I can type out my feelings to you guys. It would be nice to get a few comments every now and then guys =P. 

I love you all, and I miss you. I will be home shortly, and then I'll have so many more stories to tell you. 

Cooking!

Here's my latest cultural post about cooking! As I have said before, the food here is very inexpensive in comparison to in the states... well everything but meat and good cheese. There are also a lot of amazing places that people can get a good meal from for less than 4-5 USD; however, the hard part about going out for me though is, of course, my gluten-intolerance. I printed out an amazingly helpful little card from CeliacTravel.com that explains, in Hungarian (you can choose any language), all of the things that I cannot eat and the precautions that must be made in preparing my food. I cannot explain to you how much easier this card has made my life. Even though I can find food at just about every place that I go to, it comes with a large price tag. Most of the inexpensive, although very tasty and fairly healthy meals are made with some type of gluten product. Because of this, I often have to pay double what my friends do in order to get a meal that fits my diet. Because of this, I have resorted to cooking at home a lot more than most of my friends do. 

I have quite a lot of experience cooking. Growing up, I always loved to bake and any meal that required most of its cooking to be done in the oven was a favorite of mine to cook. Because of this, I became pretty good at making things like baked mostaccioli, meatloaf, enchaladas, etc. Upon entering college, I began practicing more with stove top cooking and I improved slowly. My pass of improvement skyrocketed this summer though when I spent the time living with my older sister Ashley and her boyfriend Ryan in their house by Notre Dame. There they cooked almost daily and I tried to help and contribute as much as I could. Because of this, I can now handle cooking most meals on a stove top with little to no problem. This has really been helpful while cooking in Hungary, but not nearly as helpful as I would have liked...

Even though food here is much less expensive, the food is very particular to Hungary. This means that finding the ingredients for meals that I love to make is practically impossible. Yellow cheeses, refried beans, affordable beef, peanut butter, many seasonings, etc are nearly impossible to find, and when you do find them they come with a steep price tag. Because of this, I have had to mix up my cooking style a little bit, and lately I have been trying to cook healthier meals so that I don't gain too much weight eating the amazing food here. My most recent cooked meal was a pasta dish with beef, spinach, and tomatoes. It kinda reminded me of a dish that my mom used to make when I was in highschool. I've also made other dishes here like meatloaf, taco dip (so hard to find the ingredients), mostaccioli, and cheesy potatoes. You can see images of the meals I've made below.

My most recent masterpiece. Pasta with beef, spinach, and tomatoes.
 Topped with Gouda cheese 

A simple breakfast: corn flakes with nectarines

A stoup I made with potatoes, rice,
summer sausage, and carrots

My taco dip. Looking all fancy too

Peanut butter and Jelly! I made peanut butter and jelly!

My bounty after a shopping
 trip to the big Tesco in the mall. 

So, there is my blog on my cooking here in Budapest. For a more general update on how I'm doing check out my next post titled "Life on the Danube"