Chapter 22 deals with a paradox
that relates transitivity and a group of paradoxes that originate from the
principle of indifference. Transitivity is a relation such as if a=b and b=c,
then a must be equal to c. Bradley Efron, a statistician from Stanford University,
developed nontransitive dice. These dice violate transitivity and with any of
these dice a betting game can be formed that is contrary to intuition.
(Efron’s first set of nontransitive dice)
The four dice are numbered so that
the winner has the maximum advantage. Lets say you and an opponent both select
a single die. If your opponent picks one first then you have the remaining
three to pick from. Both dies are tossed and the person who roles the highest
number wins. You come to the conclusion that because your opponent picked first
he has an unfair advantage. Regardless of the die your opponent picks you can
always pick a die that has a 2/3 probability of winning. This paradox violates
common sense and results from the assumption that the relation “more likely to
win” must be transitive between the dice.
The red die beats the green die, the green die beats the blue die, the
blue die beats the purple die, and the purple die beats the red die. The probability of winning with the indicated
pairs is then 2/3 because there are 36 possible throws on each pair and 24
possible outcomes on witch the first die (from right to left) beats the second
die. Efron writes that 2/3 is the greatest possible advantage that can be achieved
with his four dice.
The principle of indifference is the classical
probability theory that drives from the 18th century. “The principle states that there must be no known
reason for preferring one of a set of alternatives to any other.” (Keynes, 53)
Suppose you have a six-sided die with six different numbers on it. The assumed
probability of rolling on of those six numbers would be 1/6. This analysis rests on practical view. Now suppose you are given a shuffled deck of
four cards, two black and two red. What would be the probability of picking two
of the same colored cards? Most people
would think the probability to be 1/2. This is wrong. In fact the probability
will be 1/3 because there are 24 possibilities and 8 cases in which there are
matching cards. There are more confusing paradoxes than this one out there. The
more confusing paradoxes lead to logical contradiction.
One of the more confusing contradictions is given by
Sir Harold Jefferys in his book “The Theory of Probability”. The probability
paradox goes as follows; suppose there are two boxes. The first box containing
one black ball and one white and the second containing two black balls and one
white ball. What is the probability of picking a white ball? “There are five balls, two if which are
white. Therefore according to the definition, the probability must be 2/5. But
most statistical writers, including, I think, most of those that professedly
accept the definition, would give 1/2 X 1/2 + 1/2 X 1/3= 5/12” (Jeffrys, 370)
The actual probability contradicts most peoples hypothesis of what the
probability was going to be.
Pascal misused the principle of indifference in his
book “Pensées”. Thought 233 is worth quoting:
““God is, or he is not.” To which side shall we
incline? Reason can determine nothing about it. There is an infinite gulf fixes
between us. A game is playing at the extremity of this infinite distance in
which heads or tails may turn up. What ill you wager? There is no reason for
backing either one or the other, you cannot reasonably argue in favor of
either…
Yes, but you must wager… which will you choose? ...
Let us weigh the gain and the lose in choosing “heads” that God is… If you
gain, you gain all. If you lose, you lose nothing. Wager, then, unhesitatingly
that he is.” (Pascal, 66)
Pascal appeals the principle of indifference to a
situation that has nothing to do with mathematics. Pascal is not the only one
to make this ludicrous relation and he won’t be the last.
This
chapter was interesting because the probabilities given were not expected. The
problems given seemed easy at first but ended up being harder than I first
thought. I enjoyed the chapter a lot and could relate to it. It taught me to
think more deeply into a problem because there is probably more to the problem
than you thought.
Keynes, John Maynard. (1921) A treatise on probability,
Macmillan and Co.,
Jeffreys, Harold. Theory Of Probability. Oxford: Clarendon
Press, 1961. Print.
Pascal, Blaise,Trotter, W. F. ([1958) PenséesNew York : E.P.
Dutton,
This chapter was very interesting in the sense that it shows how an easy problem could have such a difficult idea behind it. You did a great job summarizing this and the sources you referred to were very relevant to the topic.
ReplyDeleteI found this chapter interesting mainly because I was quite confused at first. The idea with the dices took me awhile to understand. Yet I know that the whole topic of "probability" is confusing which leads to confusing examples such as the dice one.
ReplyDeleteI think this chapter was quite confusing but very compelling. I found it confusing how one dice is greater than the next one but yet the next one is greater than the one following and it continues back to the first dice. The last one ends up being greater in amount than the first one, which started as being with the maximum advantage. I continued reading and realized that the game will either be fair or bias to the first player, whether they find the die that will beat the opponent based on probability.
ReplyDeleteLike most people, I was also thrown off by the dice concept of this problem. I had never heard the word dice before, so it took me a few re-reads to finally understand it. This summary also made it a little easier to wrap my head around the concept.
ReplyDeleteThat was really interesting because it was so confusing that it made it interesting and fun to read and learn about! good job
ReplyDeleteWhile reading this and seeing the probabilities I was thrown off at first but then re-reading them made more sense. It was confusing at times but that's what probability is as the problem looked easy but behind it was a more tougher idea to grasp.
ReplyDeleteI thought this chapter was pretty cool to read. I 'm taking statistics, so reading about the probabilities was a refresher a little bit easier to understand than if I didn't have any background knowledge. I think you wrapped up a very good summary for this chapter.
ReplyDeleteThis chapter also seemed hard to understand for me because of the dice problems and its situation. But you had a really good summary for this chapter.
ReplyDeleteI thought you did a really fantastic job on summarizing a chapter that was hard to follow at times. I really enjoyed how you essentially answered your questions with the wrong answers that most people would think as correct only to give the actual answer with the reasoning behind it. What I enjoyed most was the fact that you added the quote about god from page 66 showing the impossibility of arguments on such a touchy subject. Kudos to you!
ReplyDelete