Showing posts with label Drunkards Walk. Show all posts
Showing posts with label Drunkards Walk. Show all posts

Reading Update:










Chapter 4
Chapter 4 is more exciting than the past three and, though it covers a few principals found in most elementary statistics classes, it brings in new characters to the story like Galileo and Pascal! Here goes:

  • The simple space technique (using a space or coordinates to denote potential outcomes) can only be used when the outcomes are equally probable
  • The chances of an event depend on the number of ways in which it can occur. This is important, so how do you calculate it?
  • Pascal's Triangle
  • Mathematical expectation = probability of each outcome * payoff of each respective outcome, i.e. the cost of something, say a parking meter, is the probability of each outcome ($40.00 ticket 1 out of 20 times I use one and $0.25 on the rest of the time [19 out of 20]) so $40*1/20 + $0.25 each time I use it = $2.25. So the real cost of paying a parking meter is about $2.25 each time I use one if I average it out.
  • This idea has been helpful for winning lottery tickets. For example, if you calculate the expectation of buying all of the different combinations of six numbers from 1 to 44, and compare that with the payoff of about 27 million, you will find something quite interesting! There are 7,059,052 ways to choose those six numbers with a pot of that size, each ticket is worth almost $4. Now remember that it is possible for other people to win at the same time. Appropriate the probabilities of winning alone or with other people and add them all up to come to a grand total worth of $3.31. The price to buy one is $1. What is the appropriate course of action here? Hire a whole bunch of investors, fill out 1.4 million slips by hand (each with 5 games) and coordinate a massive ticket purchasing campaign. The result? 27 million dollars! 


Reading Update:

 The Drunkards Walk by Leonard Mlodinow

 







Chpater 3

Chapter three covers quite a bit of history and a few specific examples. Here it is in summary:

 

  • Gerolamo Cardano existed and, more importantly, pioneered the idea that possible outcomes of a situation could be thought of as points in space (or coordinates)
  • This chapter covers three probability problems:
  1. You flip two coins, what is the probability that at both will heads? The answer is 25%. There are four possible outcomes of the flip: (heads, heads), (heads, tails), (tails, heads), and (tails, tails). There is a 25% chance that both or neither coins will be heads and a 50% chance that just one will be heads. The other question addressed is this: Assuming one coin will be heads, what is the probability of both coins being heads? Is it 50%? No. The condition that one of the coins will be heads only removes one of our four possible results (tails, tails). The statement does not state explicitly which coin is heads, so it could either be (heads, tails), (tails, heads), or (heads, heads). Therefore, the chances that both will be heads, knowing one of them will for sure be heads, will be 1 in 3, or about 33%.
  2. The next problem is a joke I read last week involving some college students that missed an exam due to returning late from a road trip. They appealed to their professor claiming they had a flat tire. The professor issued the exams in different rooms and, in an attempt to catch the students lying, wrote the question, “which tire was flat?" on the second page of the exam. What is the probability that they answer the same tire? The correct answer is one in four. This is because there are four tires for them to choose. Check out the table. The chance that they choose the same tire is 4 in 16, or 1 in 4.
  3. Now for the most fun one, the Monty Hall problem. This one has been around for some time and has caused quite a bit of humiliation for many many Ph.D.s as you will see later. Here is the problem as submitted by Craig Whitaker to Marilyn von Savant's column in PARADE magazine: 

Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?

The answer is yes! I will not take the space to explain why, but you can see this -- and the humiliation of numerous mathematicians-- on Marylin's website here. 

And that is about it for chapter 3. I will keep you up to date with what else I learn in this book


NOW USING CHROME

Reading Update:
The Drunkards Walk by Leonard Mlodinow









I am up to chapter three and I have learned very little and read a twenty page review of the first quarter of a statistics class I took. Here are the first two chapters in summary.

Chapter 1:
  • The right side of your brain is 'intuitive' and always tries to find heuristics
  • The left side of your brain is 'logical' and always tries to find patterns
  • If two events are occuring randomly and event 1 occurs thrice as much as event 2, in guessing which event will occur next, you will be correct more often if you simply choose the event that occurs more often everytime.
  • Praising a person for good behavior is more effective than punishing people for bad behavior
  • This can seem counter intuitive due to regression toward the mean; that is, after someone does much better than they usually do, it is natural for them to do about as well as they had been doing before that accidental success, the same goes for accidental mistakes.
Chapter 2:
  • This is a review of probability rules
  • The probability that two events will occur can never be greater than the probability that each event will occur individually
  • You can predict the probability that two events will occur by taking the product of the probabilities of the two events ONLY IF the two events are completely independent of each other.
  • The probability that, given an event with multiple different and possible outcomes, the probability that one outcome or another outcome will be the result is the sum of the probabilities of two potential outcomes (assuming that the sum of all of the possible outcomes is 1 [100%])
  • When making a point about the potential outcomes of an event, make sure you are looking at the correct proability.