Thursday, October 22, 2015

Chapter 21: Probability and Ambiguity

Charlotte Jackson
Blog Post 2
Chapter 21

                  This chapter discusses both probability and ambiguity in relation to recreational mathematics. It begins with Charles Sanders’ realization that probability is the easiest piece of mathematics to make mistakes in. For instance, if you flip a coin and I have long run on the tails side, the likelihood of flipping the coin on heads does not increase. The coin flipping myth links directly to breaking a stick into three pieces and having those pieces be able to make a triangle. With problems like these it is extremely easy for mathematicians to make blunders and perceive probability incorrectly because of its ambiguity.
                  Following the introduction is a piece on chords of a circle and an ambiguous riddle that is called “The Prisoner’s Paradox”. Essentially, the writer goes on to explain three scenarios and the probability in each that that any random chord is longer than a side of the equilateral triangle that is inscribed inside of the circle. The book explains three examples of how to prove whether or not a chord is longer than a side of an inscribed equilateral triangle. The second option offers a scenario that is most likely to have the chord be longer than the side, but the third option presents a scenario that seems most likely to occur naturally because of how random the chords are presented. In total, these three proofs prove the probability that a random chord is longer than a side of inscribed equilateral triangle nicely.
                  After the chords and the inscribed triangles are mentioned Brown goes on to explain a paradox, The Prisoner’s Paradox, that uses both probability and ambiguity. The chapter from the book will be able to describe The Prisoner’s Paradox with more detail than I. Basically, if 3 prisoners’ names are put into a hat, the name pulled from the hat is released from prison, and two of them discover that they are still in the running, then do their chances increase? In the end one of their chances remains at about 33% while the other prisoner’s odds is 66%. The two main topics discussed in chapter 21 and the chords and The Prisoner’s Paradox. All in all, all of the information that was utilized in this chapter made sense and was relatively straight forward.
                  The first article chosen out of the bibliography is L. Gillman’s, “The Car and the Goats” article which once again was all about probabilities and different formulas to reason through different situations. Game two: Marilyn’s Solution was mentioned in the chapter, but as The Prisoner’s Paradox. I am glad to see that this scenario was mentioned in chapter 21. It was very important that the writer in The Colossal Book of Mathematics explained this branch of probability. Something that I would like to see become part of the article is Baye’s Formula. This formula relates to productive probabilities, and provides an explanation for basic rules of probability.
                   Secondly I picked Georges, “Generalizing Monty’s Dilemma”. In the section of this book that was contributed to chapter 21, Georges discusses probability. Something that was mentioned that is extremely relevant to the text is how ambiguous probability is in general. It is hard to create laws of probability because at the end of the day you can never really know if a coin is going to land on heads or tails when flipped. Something I wish was discussed in more detail was Betrand Russell’s thoughts on probability. I found it very interesting that it’s impossible to know whether or not a coin will land on heads or tails with one flip, but after one million flips it can be predicted that the ratio will be 1:1. All in all I found this piece of reading to be interesting and a necessary addition to chapter 21.
                  Lastly, S. Ichikawa’s article, “Erroneous Beliefs in Estimating Posterior Probability”, is an extremely descriptive explanation for The Prisoner Paradox. As stated in the summary the prisoner paradox was discussed in the chapter. The prisoner paradox was really vital to the chapter because although it may have seemed like a silly riddle it gives an excellent description for a complex rule in probability. I would have liked to see the chapter go into further detail about how the ambiguity of The Prisoner Paradox creates more complexity. Overall, it was an intriguing read.

                  Throughout the chapter it becomes evident that probability and ambiguity go hand in hand. Probability cannot come with its own sets of rules or laws because there is no way to always accurately predict what is going to occur. Ambiguity is what makes probability more complex, but after things become less ambiguous, such as in The Prisoner Paradox, those who study it can better understand it.  

4 comments:

  1. Probability had always been something I struggled with in math, there was always a way more complicated explanation for probability, that is why this chapter was a bit difficult for me to understand at first. The first part of the chapter talking about flipping a coin was easy to understand and a discussion I have always heard in the relation to probability. The prisoners paradox was a very confusing topic in the chapter when I was reading, but Charlotte did a great job of explaining that into more simpler terms in my opinion. The chapter did not help me gain much more understanding in probability, in fact, it did confuse me a bit more. I thought this blog post did a great job of making things simpler and easier to understand however. Another interesting fact I learned that I had never put together before was how close probability and ambiguity were together. To be quite honest I had never heard them together before and once again the coin example illustrated why these two things go hand in hand. Ambiguity in my opinion seems to be what makes probability so much harder to understand. Charlotte seemed to have a good grasp on the other three sources which was a good and I believe she did a good job analyzing the chapter as a whole, and she helped me understand it a bit better.

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  2. I have always liked probability until this chapter. That is mostly because I have never dealt with ambiguity, which I have no understanding of. The first problem about the chords was very interesting because it was so random to me, but it really laid out the foundation of this chapter. I understood all the problems except the two children (at least one is a boy) and the 3 prisoners one. The 3 prisoners one was the harder of the two. I have read this over and over for a good 15 minutes just on this problem and come to any answer. I just can not understand why prisoner C has twice the chances of being pardoned than prisoner A, but I guess I just have to study it more. After this chapter and reading this blog post I now know what ambiguity is, and I now see how it is related to probability. It is a hard topic to grasp at first but i think if you look at involving probability it makes it a lot easier to understand. I enjoyed this chapter for the most part and everything i I had trouble with (except the prisoners problem) I figured it out thanks to this blog post. I thought this was a well written and explained blog post.

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  3. When I first saw the title of the chapter, I figured it would be easier to understand because I am well aware of probability and its functions. But it was actually more difficult than I thought. The coin flip example is one of the examples most frequently used when describing probability. I agree with Frank that the example of the chords was randomly placed compared to the rest of chapter. The second part of the prisoner’s paradox I thought was a good connection to probability and ambiguity. How Gardner explained it I think was confusing and Charlotte did a better job of explaining it than how it was described in the chapter. I agree with Tiffany that the chapter did not help me learn more about probability that I didn’t already know and I think that could be similarly said about other readers. Although that could be a good thing and the chapter could be understood by more people. But if Gardner’s goal of the chapter was to teach readers about probability and ambiguity than in my opinion, he missed the mark on that. Charlotte did a good job of explaining the chapter and the references as well. She really seemed to understand the chapter and how the references play a part in the whole explanation.

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  4. I wanted to read this chapter because of the amount of straightforwardness associated with probability. One of the examples in this chapter I really liked was the one where you break a stick into three random sized pieces and make a triangle with them. If the stick is broken up randomly then the chance that we are able to make a triangle is 1/4. The geometrical analogy for this example I found was kind of confusing but extremely interesting, I would never have been able to make a proof for the stick problem the way they did with the equilateral triangle. Another part of this chapter I really enjoyed was the prisoner paradox. In my opinion Charolette did a very good job of explaining this portion of the chapter. I enjoyed the examples of the procedures to find the probability of this to be a bit strange but very interesting. I would never have imagined that for the same problem you can get different probabilities for the same answer, I guess that explains why at the beginning of the chapter Gardner says probability is so easy for experts to "blunder" about. All in all I really enjoyed this chapter and Charolettes summary of it, however I felt this chapter was too short and there could have been more interesting examples in it.

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