I spoke about the Problem of Points, which was puzzling people as long as 500 years ago. On pages 32-34 of
Grimstead and Snell you can read translations of the letters exchanged by Fermat and Pascal about this problem in 1654.
G&S also write:
The problem had been a standard problem in mathematical texts; it appeared in Fra Luca Paccioli’s book summa de Arithmetica, Geometria, Proportioni et Proportionalita, printed in Venice in 1494 in the form:
I mentioned classical, frequentist, and subjective approaches to probability. There are others: such as the propensity approach. Indeed, the question, "What is Probability?" can take us into the realm of philosophy. If I roll a die and do not show you the answer, what is the probability it is a six? Some people might say it is 1/6. Others might say it is 1 or 0, because it either is a six or not. (If it appeals to you to delve into philosophy, then have a look at this article on Probability interpretations.) Fortunately, in our course we work within the context of some mathematical axioms and therefore have a well-defined approach that does not depend on interpretation.
The arcsine law that I mentioned today is the second of the three laws described here. The other arcsine laws concern the proportion of the time that the random walk is positive, and the time at which the random walk achieves its maximum. My purpose in including this in our first lecture was to show you something that is surprising and which is proved in a clever way (by the reflection principle and matching paths in a 1-1 manner).
There is a space after each post for you to write comments or ask questions using the IntenseDebate commenting system. Please feel free to do so. Other students will probably benefit.
The problem had been a standard problem in mathematical texts; it appeared in Fra Luca Paccioli’s book summa de Arithmetica, Geometria, Proportioni et Proportionalita, printed in Venice in 1494 in the form:
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"A team plays ball such that a total of 60 points are required to win the
game, and each inning counts 10 points. The stakes are 10 ducats. By
some incident they cannot finish the game and one side has 50 points
and the other 20. One wants to know what share of the prize money
belongs to each side. In this case I have found that opinions differ from
one to another but all seem to me insufficient in their arguments, but I
shall state the truth and give the correct way."
I mentioned classical, frequentist, and subjective approaches to probability. There are others: such as the propensity approach. Indeed, the question, "What is Probability?" can take us into the realm of philosophy. If I roll a die and do not show you the answer, what is the probability it is a six? Some people might say it is 1/6. Others might say it is 1 or 0, because it either is a six or not. (If it appeals to you to delve into philosophy, then have a look at this article on Probability interpretations.) Fortunately, in our course we work within the context of some mathematical axioms and therefore have a well-defined approach that does not depend on interpretation.
The arcsine law that I mentioned today is the second of the three laws described here. The other arcsine laws concern the proportion of the time that the random walk is positive, and the time at which the random walk achieves its maximum. My purpose in including this in our first lecture was to show you something that is surprising and which is proved in a clever way (by the reflection principle and matching paths in a 1-1 manner).
There is a space after each post for you to write comments or ask questions using the IntenseDebate commenting system. Please feel free to do so. Other students will probably benefit.
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Charlie · 585 weeks ago
a) you toss a biased coin { P(h) = 1/3, P(t) = 2/3 } 4 times. What is the probability of observing { h,h,h,h } ?
b) you roll a fair 3 sided die { P(1)=1/3, P(2)=1/3, P(3)=1/3 } 4 times. What is the probability of observing { 1,1,1,1 } ?
Via the lecture definition b) is an exercise in classical probability. Is a) ?
Richard Weber 35p · 585 weeks ago
A more satisfactory way to treat a) is via the set of probability axioms that we will come to in Lecture 4.
Guest01 · 584 weeks ago
On page 3 of the notes, you write "let $B_k = A_k - A_{k-1}$ "
Should this not be "let $B_k = A_k / A_{k-1}$ " and hence " $|B_k| = |A_k| - |A_{k-1}|$ " ?
Regards.
Richard Weber 35p · 584 weeks ago
Guest01 · 584 weeks ago
Thank you for your reply. :)
Richard Weber 35p · 584 weeks ago