Calculus junkies

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For the calculus junkies:

The longer an inevitable occurrence does not happen, the higher the probability of it happening becomes.
Ok
While the existing pattern may cause complacency.*
All humans may comprehend this.
The sun may rise in the morning, but it will not rise every day forever. The Earth will not support life forever. (See LaPlace')
This is not a probability, but a certainty.

That is untrue. If the occurrence is inevitable, then the probability of its occurrence is 100%. A probability cannot be higher than 100%, so the probability of the occurrence of this event cannot possibly increase.

Furthermore, I fail to see what your post has to do with calculus.

Ditto.

1. I believe she is saying the *probability per day* increases, since it is inevitable. For example, some astronomers say the Sun will inevitably die, and the life on Earth will consequently die thereafter, however, the probability it will happen tomorrow is not 100%.

2. She might mean that calculus is needed to determine such probabilities. In this problem we would create a distribution function to represent the per day probability of the Sun going out, then integrate that function to determine probabilities. The probability on a probability distribution does increase as the days increase, and inevitably approach 1 as the days = infinity.

You are right as far either the probability of the event happening at some unspecified time is concerned, or if you allow an infinitely long future. If the time span in which the event must happen is finite, and if you calculate the probability per time, for example the probability of the event happening per second, then starkid is right.

Example: put one bullet into a revolver, spin it, pull the trigger once per second. It is inevitable that you will eventually hit the chamber with the bullet. Every time you fail to do so, the probability of you firing the bullet in the next second increases.

Don't know what it has to do with calculus.

How would we know the probability per day without knowing how much time is left before the event occurs? For example, if we know that the event will occur sometime within the next week, the probability of it occurring today would be 1/7, the next day, 1/6, and so on. Using the OP's example, no one has a precise idea of how many days are left until the sun "stops rising," so we'd have nothing on which to base our probabilities.


Integration is only meaningful for continuous functions. If we are to represent probabilities per day, then wouldn't we have a discrete function, and be obliged to use summation?

Lolwut.

Laplaces' argument was from ignorance. Bayesian priors can be reconstructed on the basis of assumed equal probability. The probability of the sun rising tomorrow then, according to Laplace is not, indeed, 100% (i.e. all observations have seen this but), to be more precise:

n*s+1
/n*s+2 i.e. the number of times the sun has risen plus the two 'unobserved' events of the sun rising and the sun not rising.

This has nothing to do with stellar evolution and the process whereby a sun-like star leaves the main sequence. As it ceases to burn hydrogen in the core contracts to a denser state due to helium fusion, develops a shell of hydrogen fusion. And thus it's temperature gradient, and thus luminosity increases, causing an increase in solar radiation and an expansion of its outer layers to a somewhat gigantic size (perhaps 2 AU).

You have to represent the event happening in a day with a probability function. You have to use whatever data / assumptions you need to develop said function.

In college, I did a lot of "polynomial curve fitting" where data was plotted on a graph, and then using "curve fitting" means, a function was developed to represent the data, and then that function was used as a predictor of likely behavior per any input (i.e., any future day in this case).

In this case, you would have to consult astronomy people, and look at advanced mathematics to see how to develop said function.



Sure. However, once we have the probability function from above, then we can see the probability of occurrence per day.



Sure.

There is no way to know unless we determine the function as the function could be a discrete function or continuous function depending on how we model it.

However, I would think it would be continuous. Over infinite days, it would be assumed that there is no day where such an event becomes non-probable, and the function would not have to have a break i.e. "separate", rather it would follow some continuous path.

If there is such a day, or some assumption, then we might be able to represent the probability function as a piecewise-defined function, and break up the computation by summing the integration of each piece of the piecewise function.

If any astronomers say that, then they should be summarily fired from their jobs for being incompetent.

All life on Earth will die long of years before the sun dies.

But, astronomers not being specialists in biology, would lack of that knowledge really imply incompetence?

And, now that I've noticed it, what do we even mean by "die" in reference to the sun?

Yes it would imply incompetence. I can't imagine any legitimate astronomer not being at least aware of the events that will happen as the Sun runs out of hydrogen to use to fuel its nuclear fusion.

When the Sun runs out of hydrogen, it's not just going to wander away or shut down. Instead, it will expand in size and make the surface of the Earth inhospitable for life as we know it from heat and radiation. It is expected to eventually grow until its diameter is at least as large as the orbit of Venus and maybe until it is as large as the diameter of the Earth's orbit. And the Sun will likely still be around for a few billion years after that.

The astronomical events, sure, but the biological events? For all life, even viruses and bacteria?


This is what you mean by "die"?

Not me.

I would say the the Sun dies when it cools off and no longer shines.

Running out of hydrogen isn't even the mid-point of the life of the Sun.

How many viruses and bacteria will be able to survive billions of years of heat near or higher than that of boiling water?

But wouldn't it continually cool off after running out of hydrogen? And what marks the end of shining?



I've no idea, and I don't expect astronomers to know either. I was merely wondering if you expected them to know such things.