Half a coin is stranger than none
The physicists Dirac and Feynman, both bold when it came to new mathematical ideas, both said we should think about negative probabilities.
What would it mean to say something had a negative chance of happening?
I haven't seen many attempts to make sense of this idea... or even work with this idea. Sometimes in math it's good to temporarily put aside making sense of ideas and just see if you can develop rules to consistently work with them. For example: the square root of -1. People had to get good at using it before they understood what it really was: a rotation by a quarter turn in the plane.
Here's an interesting attempt to work with negative probabilities:
• Gábor J. Székely, Half of a coin: negative probabilities, Wilmott Magazine (July 2005), 66–68, available at http://www.wilmott.com/pdfs/100609_gjs.pdf.
He uses rigorous mathematics to study something that sounds absurd: half a coin. Suppose you make a bet with an ordinary fair coin, where you get 1 dollar if it comes up heads and 0 dollars if it comes up tails. Next, suppose you want this bet to be the same as making two bets involving two separate 'half coins'. Then you can do it if a half coin has infinitely many sides numbered 0,1,2,3, etc., and you win n dollars when side number n comes up....
... and if the probability of side n coming up obeys a special formula...
and if this probability can be negative whenever n is even!
This seems very bizarre, but the math is solid, even if the problem of interpreting it may drive you insane.
By the way, it's worth remembering that for a long time mathematicians believed that negative numbers made no sense. As late as 1758 the British mathematician Francis Maseres claimed that negative numbers
"... darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple."
So opinions on these things can change.
By the way: experts on probability theory will like Székely's use of 'probability generating functions'. Experts on generating functions and combinatorics will like how the probabilities for the different sides of the half-coin coming up involve the Catalan numbers.
The physicists Dirac and Feynman, both bold when it came to new mathematical ideas, both said we should think about negative probabilities.
What would it mean to say something had a negative chance of happening?
I haven't seen many attempts to make sense of this idea... or even work with this idea. Sometimes in math it's good to temporarily put aside making sense of ideas and just see if you can develop rules to consistently work with them. For example: the square root of -1. People had to get good at using it before they understood what it really was: a rotation by a quarter turn in the plane.
Here's an interesting attempt to work with negative probabilities:
• Gábor J. Székely, Half of a coin: negative probabilities, Wilmott Magazine (July 2005), 66–68, available at http://www.wilmott.com/pdfs/100609_gjs.pdf.
He uses rigorous mathematics to study something that sounds absurd: half a coin. Suppose you make a bet with an ordinary fair coin, where you get 1 dollar if it comes up heads and 0 dollars if it comes up tails. Next, suppose you want this bet to be the same as making two bets involving two separate 'half coins'. Then you can do it if a half coin has infinitely many sides numbered 0,1,2,3, etc., and you win n dollars when side number n comes up....
... and if the probability of side n coming up obeys a special formula...
and if this probability can be negative whenever n is even!
This seems very bizarre, but the math is solid, even if the problem of interpreting it may drive you insane.
By the way, it's worth remembering that for a long time mathematicians believed that negative numbers made no sense. As late as 1758 the British mathematician Francis Maseres claimed that negative numbers
"... darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple."
So opinions on these things can change.
By the way: experts on probability theory will like Székely's use of 'probability generating functions'. Experts on generating functions and combinatorics will like how the probabilities for the different sides of the half-coin coming up involve the Catalan numbers.
