Odds vs Probabilities in extreme cases

If science were to calculate something at 100 billion to one odds how would that affect the probabilities when compared to something with say 3 to 1 odds.

It would be 33.3 repeating billion times less likely. Or effectively a chance of zero. But I think I’m not understanding the question.

I’ve read this sentence an even dozen times and I still can’t parse the intent. The probability of a single event are the “odds”, i.e. the betting ratio except defined as a likelihood i.e. 3:1 odds is p=0.75=(3/(3+1)).

Stranger

The number given in odds isn’t quite the same as probability, as @Stranger_On_A_Train mentioned. 3 (as in 3 to 1 odds) is significantly different from 4 (as in a 1 in 4 chance). But once you get to 100 billion, that +1 will never matter (especially since the “100 billion” figure is almost certainly rounded).

I.e., if we had perfect precision, then something with 100 billion to 1 odds would have a probability of 1/100,000,000,001, rather than the 1/100,000,000,000 one might expect… but those are so close that it doesn’t matter.

I think my understanding of the word probabilities was flawed.

I thought that might be the case, but what are the odds.

The bookmaker can set the odds to whatever he or she wants. At 3-to-1 odds, you stand to make a $3 profit on a$1 bet. This is a fair game if the underlying probability equals 1/4:

\left(\frac{1}{4}\cdot 3\right)+\left(\frac{3}{4}\cdot -1\right) = 0.

One out of 4 times you win $3, but this is balanced by losing$1 three out of 4 times.

You ask how this changes with 100 billion to 1 odds. Well, since the bookie does not have \$100 billion dollars, you would be getting scammed. You are not getting such odds, though: e.g. in October 2017 paddypower.com had the odds of the world ending at 500/1 (slashed to 100/1 after Trump started talking to Kim Jong-Un), and 4/7 for Kim remaining Supreme Leader beyond 2031. No billions, even if you cash in on the end of the world!

During my Handicapping the horses’ days I subconsciously created my own meaning for probabilities which really amounted to nothing more than changing the values of different factors often used when handicapping a horse or applying factors not usually applied.

Well, you can certainly make the distinction between true probabilities, which a handicapper estimates, and the implied probabilities derived from the odds. As a punter, if I make a bet at 3/1 against, that’s an implied probability of 0.25, the simple mathematical relationship described above. If the bookie is offering that price, in his expert judgment the true probability of the outcome is less than 0.25 (setting aside weight of money effects that might lead the bookie to offer a better price). If you look at a bookie’s prices and add up the implied probabilities from the odds offered for the entire field in a race, they will add up to more than 1.

“Probability” is supposed to mean something, though. For example, suppose you have in mind a model for the outcome of an experiment, given some hypothesis (your different factors \theta). One possible approach, if you have “your own meaning” in mind, is to incorporate your subjective probability as a prior probability \pi(\theta). Then, given observations x, you can apply Bayes’s theorem in the form

p(\theta \mid x) = \frac{P(x\mid\theta)\pi(\theta)}{\int P(x\mid\theta')\pi(\theta')\,d\theta'}

to derive a posterior probability density function.

Without getting into the technical details, the statistics of this function p(\theta\mid x) tell you something about the likely value of the parameters given your observation.

Do you mean something like a real-world version of:

Captain James T. Kirk: What would you say the odds are on our getting out of here?

Mr. Spock : Difficult to be precise, Captain. I should say, approximately 7,824.7 to 1.

Kirk: Difficult to be precise? 7,824 to 1?

And yet…they succeeded. So despite the high odds, the actual probability was closer to…1.

Like the odds of winning the lottery are YUGE, yet…people do win.

It goes to my theory of the “one-time event” like Kirk and Spock’s adventure’s probability being 50-50. Spock and Kirk either would make it, or not. No one would be alive at the end to argue the odds were actually 5690.4 to 1, or ten million to 1.

In “reality”, of course, Spock is misusing probability. To create such precise odds, you’d need a data base of hundreds of attempts of people trying to do similar commando raids, with a factor for the personalities involved, weighted by training and previous attempts of similar tasks and their survivability rates. I’m just not sure if the writers are ignorant, or Spock is. Because, given the history of these two, the odds are actually 100% success.

Nothing gets me more hot and bothered than Bayesian statistics. Please keep going.

Stranger

I was thinking of the probability of life occurring on our planet. Impossible to calculate but if you could factor in everything it would be 1

Thinking back to my pony days I gave very heavy weight to horses that had good times but bad records and very recently made a switch to a winning trainer, owner or jockey, usually all 3. If I could find horses like that going off at 10 to 1 or better I would pick them and had more than my share of winners on just that.

There is no “probability of life occurring on our planet”, unless by that you mean a new form of life unconnected with existing DNA-based organisms. Probabilities describe the likelihood or expectation of future occurrence of some phenomenon. You can look back on a population of events and develop a posterior probability, but that is just a modification of your previous expectation given the frequency of observed occurrence as applied to predictions of future events.

Stranger

That’s just the thing. Unless you can replicate the experiment, like @Just_Asking_Questions says, if your experiment is, e.g., a single flip of a coin which comes up “heads” with probability p, and it came up heads, and for the sake of argument let’s say that before the flip you had no idea what the probability was, afterwards you know p=1 is more likely than any other value, but you do not know that it was not really 0.666.

I don’t think it’s correct in principle to say he’s misusing probability. We can use probability to express what we do know about an outcome, while acknowledging that our ignorance of many factors creates uncertainty that is functionally equivalent to true randomness. I mean, if we don’t do this, we cannot even use probability to talk about a dice roll, which is a deterministic outcome of the forces involved in the throw. We’d have to reserve probability to quantum effects that are truly random.

But where I certainly agree with you is that it’s an annoyingly common trope among writers that expressing results to an ridiculously inappropriate level of precision is a sign of how incredibly smart someone is. In reality, of course, if Spock were smart he’d acknowledge the vast uncertainty, and give a probability estimate without the false precision, or just say he didn’t know.

“The probability of making it through that asteroid field is 6.829%…plus or minus the 25% uncertainty bounds on our model.”

Stranger

You seem to be asking, out of all the planets in the universe, how likely or unlikely was life appearing on our planet?

Since we don’t know the exact answer to why or how life appeared on Earth, any attempt at an answer relies on a series of assumptions. So far all attempts have been meaningless because nobody knows what assumptions are important and what the odds on any of them are.

Somebody is going to raise the 1961 Drake Equation that tries to calculate the odds through assumptions. My personal opinion is that, colloquially, it is a star-sized load of horseshit.

I can contribute one fact to the mess. Back in 1952, Willy Ley wrote “Introduction: Other Life Than Ours” for the science fiction anthology Travelers of Space. In it, he set out the Drake Equation in almost full form.

We can reason like this: Our island universe, our galaxy, contains at least 15 billion suns. … Being as pessimistic as is consistent with good sense we’ll put the number of suns with planets down as one billion, or 1,000,000,000. Each of these can be expected to have at least two planets of the type of Earth and Mars. This gives us two billion planets in our galaxy that can be expected to harbor life.

If we say that just one out of a hundred of these planets has progressed far enough to produce intelligent life of some sort, we arrive at the fantastic figure of twenty million planets with intelligent beings. Again, if only one out of a hundred of these intelligent types have progressed as far in the engineering sciences as we have, we get two hundred thousand planets on the verge of space travel.

And if, again, one out of a hundred is no longer just “at the verge” – but here begins the realm of science fiction.

I’d give Ley the credit, but I also don’t want him bearing the blame for all the wasted hours of mindless thought that has resulted.

Never tell me the odds!

And yet…they successfully navigated the asteroid field. AND escaped the stupid asteroid monster. Wonder if ol’ goldenrod factored THAT in.

Which brings up the other point of these kinds of things, which could be called the “Challenger Fallacy”.

“Sir the odds of success are 7,824.7 to 1.”
“Last week, when we escaped the rogue computer, you said the odds were 5623.9 to 1.”
“Yes”
“And we made it. And the previous week with the planet of amazon women, you said the odds were 9356.17 to 1. And we made it. Why do you bother? I shouldn’t even listen to you. You just put doubt in my head. Let’s go!”

[they die]