Understanding Bayes Theorem With Ratios

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41

@ Ralph,
The issue here is what you are applying the probability to. Saying “every ticket has x probability of being a winner” is an error of literal expression, not an error of the numbers. That’s how its expressed, because it is short and easily understood by the buyer.

A better literal expression of the concept is:

“Every ticket holder picks 6 numbers. Every ticket holder, prior to deciding their numbers has X chance of choosing the numbers that will be correct”

OR

more accurately, “When the drawing happens, there is X percent chance they will draw the numbers you picked on your ticket”

If we assume it is a one number per person ticket type lottery, than there is STILL equal chance per ticket. Yes, only ONE ticket actually can be the winner and all the others cannot, however if there are 100 tickets, and each person gets 1, then EACH PERSON has a 1% chance of drawing the correct ticket. (Once the tickets are drawn, the odds change, because the conditions change but we do not know by how much, until the numbers are announced).

Betting odds on the other hand are NOT ODDS AT ALL. People do not understand this. Betting odds are more accurately BETTING LINES. They are not probabilities of a team winning nor are they meant to be.

Say Dallas and Washington are going to play. Washington is given 60/40 odds over Dallas. (Now, for the record, 60/40 COULD actually be expressed as a pure probability, in that many sports predictors will use computer programs to enter both teams attributes, run 1000 or so simulations of the game, and then relate that WAS won 60% of the sims)

HOWEVER, whats really happening is, the casinos are calculating expected betters vs expected payouts. If WAS and DAL both had the same betting odds, all the smart money would bet on WAS and the casinos lose money. So, by adjusting the odds to adjust for the “favored team” they entice some people to place money on the perceived loser, and reduce the amount the have to pay out to the winner, should the perceived favorite win.

THEN they adjust again based on betting pools.
Even though WAS is a better team, meaning more “logic betters” will be on them, DAL has a large fan base, meaning more “I just like them” betters will come drop money on them, SO the casinos push a few points back to DAL to offset fan bets.

Remember, at the end of the day, vegas is not setting the odds hoping they pick the winner. They are setting the odds according to how they expect the bets to be placed, so that no matter which side wins, they can use the losers to cover the winners and still collect 20%.

42

It sounds like Greene was trying to get across Boltzmann’s argument as to why thermodynamic entropy should go up. That is a bad argument, but it’s an easy trap to fall into. Given what knowledge we have now about a particular isolated system, the vast majority of possible future states have higher entropy than the present state; Greene may have demonstrated this with torn paper and the word ‘disorder’, but it’s still a true fact about thermodynamic entropy. Unfortunately, the exact same argument shows that the vast majority of possible PAST states have higher entropy than the present state! So this doesn’t explain why (thermodynamic) entropy goes up rather than down.

43

Thanks Keith, that’s a great way to put it!

44

I personally have an easier time thinking with probabilities than with odds; but I still like this idea of an evidence adjustment.
original probability * evidence adjustment = new probability

It’s easy to see this form with a slight re-write of Bayes Rule:
$$p(A)\cdot \frac{p(X|A)}{p(X|A)\cdot p(A) + p(X|\tilde{ }A)\cdot p(\tilde{ }A)}=p(A|X)$$

45

@Ralph Schneider:
I guess you make a central mistake, but I think you realized already.
If you flip a coin, the chances for head an tail are equal:
One object (the coin) has both sides (or two possible states), if you flip it,
it will either be head or tail.

Now you have 1.000.000 papers, and 1 has the winning number written on it.
From this perspective, you have also two possible states (won/not won), but
only one out of one million will satisfy the condition, thus giving the 1:1.000.000 probability.

It is obvious to say: if you have drawn the winning ticket, it is the one out
of 100% winning tickets : there was only one.

46

@Stefan Sonnenberg
I think you “make a central mistake” in not understanding my point - you have merely re-stated the obvious! My point is simply that it is obviously incorrect to state that each “piece of paper” has an EQUAL chance of containing the number, since only one has it.

47

But the word “chance” is key there. It defines the use of probability which deals in the likelihood of specific unknowns.

Ralph, nobody is saying that a piece of paper that doesn’t have the number on has a chance of becoming a piece of paper that has it. We’re saying that when choosing one of the pieces of paper, the winning one being unknown, your knowledge is not sufficient enough for you to know that any piece of paper is a better choice.

I think that you have a good point to make somewhere in your argument about the general trend for people who make decisions based on probability to believe that the decision is thus safe or guaranteed. But you seem determined to throw out the baby with the bathwater - probability is not only useful, it’s inherent in our brains’ decision making processes, and without formalizing it we would only worsen our decisions.

48

@wererogue
Far from being “determined to throw out the baby with the bathwater”, I made this precise point in a much earlier post on this thread.

If a handful of respondents bothered to read - and think clearly about - my previous posts, I would not have to repeatedly correct what they ascribe to me.

I’m not sure if there is a word for hanging an idea on a person which they do not hold and then criticizing them for that idea, but unfortunately it’s a centuries-old, infamous, common, disappointing form of intellectual dishonesty.

49

And there lies the absurdity - even if your ability does not vary, your “probability” of success is said to vary from time to time according to what OTHERS do!

50

Only when you ignore preparation, and equipment etc. There is a separate probability for “person will succeed in crossing the Grand Canyon” and “Well-prepared person will succeed in crossing the Grand Canyon”, as well as “unprepared person will succeed in crossing the Grand Canyon”.

As an aside, the knowledge gained from another would increase your probability only by putting you into a category with a better probability. The probability based on observation, if you ignored relative knowledge levels, would remain the same.

51

The trouble is, such categorizations or particularizations are endless. Did the fact that he believed in the assistance of God increase his “probability”, and/or that his father was also a tightrope walker, and/or that he tied his left shoelace before his right (ad infinitum)? So we have to look at each individual, not meaninglessly ascribe “probabilities” to individuals based on generalized observations of others along with accompanying assumptions.

52

@Ian, Ralph, Toby: Thanks for the discussion! I think this would be a good topic to clarify, it brings up the (somewhat philosophical) nature of what a probability means. Here’s my take:

A probability represents the uncertainty in the knowledge of the person making the prediction, and is based on a dramatically simplified model. It’s not a statement of fact, or a likelihood an individual element can reach some ‘success state’, unless that is allowed by the agreed-upon model.

For example: when dealing with idealized dice, we create the assumption there’s a 1/6 chance of every number appearing, and in this idealized model we assume we can eventually get every number to show up on a single die if we roll it enough (with near-certainty).

For a model with non-homogenous elements, like people crossing the Grand Canyon, probabilities represent our a-priori knowledge of “number of desired outcomes / number of attempts”. There’s almost certainly further classifications which, in a Bayesian way, act as clues to what leads to a more- or less-successful outcome. In these situations, the probability acts as a demographic guide, similar to an average or median, which essentially says “This number is representative of the attempts made by this population.” If we start adding expert crossers, etc., we aren’t changing the likelihood of success for the existing individuals, but we are changing the properties of the group.

I think Ralph’s point may be that in a heterogeneous population, a probability doesn’t really apply to an individual [they are or they aren’t something]. Ian/Toby make the argument that a metric is still useful: even if nobody has the “average” 2.3 kids, it’s useful to know that population A has a higher average than population B :).

53

A typically excellent explanation, Kalid! As you say, discussion of probability can be “somewhat philosophical”, with ancient arguments of “free will” vs. “determinism”, for example, peeping in.

54

Thanks Ralph, happy you enjoyed it. Yep, there’s definitely a few interesting rabbit-holes thinking about math can lead us down :).

55

My probability that I can cross the Grand Canyon on a tightrope depends on two things: my ability to cross the Grand Canyon on a tightrope and my knowledge about crossing the Grand Canyon on a tightrope. When somebody crosses the Grand Canyon on a tightrope, this increases my knowledge, so I can thank them for that; but it doesn’t increase my ability. Either way, however, it increases my probability.

It’s even clearer if somebody attempts to cross the Grand Canyon on a tightrope and fails. This also increases my knowledge, and so I thank them too, even though this time it decreases my probability.

56

We can’t always do that. That’s what probabilites are for - to give us some form of visibility when we don’t have all the specifics.

It seems like you’re really confused as to the role of probabilities, and it seems to me that this thread has become something that is going to be unhelpful to people coming here to learn. Perhaps if Kalid is interested, this is a sign that a broader-level article on probability would be useful?

57

>even if your ability does not vary, your “probability” of success is said to vary from time to time according to what OTHERS do!

Even if your ability does not vary, your probability of success will vary according to what knowledge you have (and that can be affected by what others do). This is perfectly reasonable if your probability depends on both your ability and your knowledge of that ability. And the idea that the only useful notion of probability (the one which isn’t always 0 or 1, only we don’t know which, which has no practical applications) is one that depends on what information one has, as we established upthread.

58

The probability of survival, based on observation, would increase minutely to represent our new knowledge that it is possible under certain circumstances to survive a 1,000 foot fall.

59

And we should all thank that fellow who recently crossed Grand Canyon on a tightrope, because at the instant he completed his crossing he immediately increased the probability of all of us to successfully do it too!

60

Of course, we can safely assume that many things are certain. For example, the chances of a naked person surviving a 1,000 foot fall onto rocks could intelligently be assumed to be 0%, BASED ON EXPERIENCE, although it’s also true that, since they would either survive or not, their chances were either 100% or 0%. But if miraculously someone did survive, the chances of everyone else surviving would not suddenly increase, although the AVERAGE number (based on experience) would increase. If on average 1 in 5 people survive a 100 foot fall, we cannot say that everyone has an equal 1 in 5 chance of surviving. If the average number of children per family is 1.27, we do not conclude that every family has 1.27 children.