Date: Feb 16, 2013 2:45 PM
Subject: Re: probability question about the dice game
>David C. Ullrich wrote:
>> >Staying with the theme of odds terminology, but moving away
>> >from the argument,
>> Giggle. Yes, now would be exactly the right time to "move away
>> from the argument". Guffaw.
>I don't find it that hilarious. Obviously, I've been shown to be
>wrong about what standard usage is.
Obvious to you perhaps, but not necessarily to the general
readership. Thus, before changing the topic, if you come to
the realization that you were wrong, you should first admit
that you were wrong, making it clear that you are retracting
your earlier claim that Ullrich was wrong.
>When bookmakers say an event is "2 to 1" without qualifiers
>like "on" or "against", they mean that the probability is
For offered odds, "against" is always assumed.
>I'm certainly surprised to hear people say that
> "The odds are 2 to 1 that it will rain" means a probability
> of 2/3.
Here's another source ...
From the text:
Elementary Probability, 2nd Ed (2003)
=== BEGIN QUOTE ===
If the occurrence of some event is denoted by A, then A^c denotes
the nonoccurrence of A, (= the event that A does not occur).
If the probability of A is p and the probability of A^c is q, then
the odds against A are q : p (pronounced q to p).
The odds on A are p : q.
=== END QUOTE ===
>I wonder why no one says that the odds of winning the lottery
>are "one to millions"
I already gave what I thought was an explanation of that. I'll
try again ...
When the probability of an event is low, it's common to express
the odds as odds _against_, rather than odds _for_. Moreover,
if the lowness of the probability is obvious, the word "against"
can be omitted without loss of clarity.
So if one says
"The odds of winning the lottery are 10 million to one."
it really means
"The odds _against_ winning the lottery are 10 million to one."
On the other hand, for betting odds, the default is always
"against", regardless of whether or not the probability is low.
Thus, if fair odds of a-to-b are offered on a given event, that
means the odds _against_ the event are a-to-b. Equivalently,
it means the (perceived) probability of the event is b/(a+b).
Thus, if a fair bet on an event is offered at 2-to-1 odds,
that equates to a probability of 1/3 that the event will occur.
Similarly, if a fair bet on an event is offered at 1-to-2 odds,
that equates to a probability of 2/3 that the event will occur.