Basic Probabilities

Factual Questions
1

I’ve been trying to teach myself something about probabilities. I’ve tried in the past and never seemed to get anywhere until I started playing poker; especially Texas Hold’em. ( My success BTW was with the probabilities and not the game itself, at which I suck.) I think my success came from the concrete numbers involved. As I try to apply myself to more abstract problems, I’m right back where I started. Any basic inout would be helpful.

One case I’ve been trying to figure out has to due with windows of opportunity.
Imagine that you have a black box with a light bulb inside. The light is on a timer such that it is on for T seconds and then turns off for an unknown period. If the box is opened and the light is on what is the probability that the light will stay on for T/2 seconds, for example? How exactly does one figure the varying probabilities? How does this relate to the standard normal curve?

Any insights would be helpful.

2

Huh?

If the the time the light bulb stays on in seconds is unknown and the time in the lightbulb stays off in seconds is unknown, how can any meaningful probability bee computed?

3

I’m assuming that you mean that we know what T is, we just don’t know how long the intervals are when the light is off. But the good news is that that doesn’t matter. Suppose that T is ten seconds. If we open the box and observe that the light is on, the probability that it will stay on for at least five seconds is exactly one half. Likewise, if we wanted to know the probability that the light will stay on for at least nine seconds after we open the bok, it would be .9 .

The only tricky part is that these probabilities only apply to cases where the light is on. In other words, suppose we opened the box one hundred times, but only wrote down how long the light lasted in cases where it was on when we opened the box, ignoring all other cases. Then half of the cases that we wrote down should have the light staying on for at least five seconds.

4

Normal is all about having enough sample. Seems to me though, if you play casino, and you play long enough, you lose.

Stocks are much better, if you lose, you only lose a percentage.

But for hold-um, I don’t know, bet a pair of queens or better big, and A K the same.

5

I’m pretty sure you meant 0.1 here.

6

Yup.

7

First let me apologize for starting this thread and then abandoning it. I’m guessing it is also a faux pas to revive it for my own selfish ends after it has been dead for about a day and a half-- assuming it was not still born.

Part of my question is-- given T=10 seconds-- is the probabiltiy of the light being on for 9 seconds equal to the probabiltiy of the light being on for 1 second?

Does it really matter what the value of T is? Isn’t the probabiltiy, in this case, a mattter of factors ( for want of a better term.) If T = 10 and the probability of the light being observed for 5 seconds is 1/2 ( .5) then, if T = 20, don’t we KNOW that the probability of the light being obsereved for 10 seconds is half as well?

I am asking… I don’t really have a grasp of applpying probabilities to real situations outside of cards and jars of marbles. ( Give me a black box with light bulb any day!)

8

If you look in the box, the odds that the light will stay on for exactly whatever number of seconds you specify is 1/(total number of seconds the timer is set for). When you start getting into "AT LEAST"s or "LESS THAN"s, all you need to do is ADD up the probabilities that will fulfill those conditions.

Here’s a chart for T=10:


Odds the light will stay on for exactly X seconds:
__X_|_Odds_
 01 | .10
 02 | .10
 03 | .10
 04 | .10
 05 | .10
 06 | .10
 07 | .10
 08 | .10
 09 | .10
 10 | .10

See, if you did this a million times and wrote down your results only when the light was on, each result should come out to nearly 10% of the total.

Now, if you want to know the odds for the light staying on for at least three seconds, you just add up the probabilities for three on up, and you’ll get .80, or an 80% chance. In other words, when you open the box and the light is on, there’s an 80% chance it will last longer than three seconds.

9

Pehaps I’m telling you something you already know here, but…

An important distinction between this problem and jars of marbles is that this is not a discrete problem – that is, you are dealing with a continuous variable (namely time).

For comparison, take a jar of 20 marbles labeled “1” through “20”. An experiment in which you pick a random marble has exactly 20 different possible outcomes, all of which share equally the total available probability of 1. That is:

Ball 1: p=1/20
Ball 2: p=1/20
…
Ball 20: p=1/20
TOTAL: p=1

In your light/box case, there is an infinite number of possible outcomes, namely T=0.000000, T=0.000000+(an infinitessimal amount), T=0.000000+(another infinitessimal amount)…, etc. The probability of any one of these outcomes occuring is zero, so with a continous variable one must ask about ranges of values instead of distinct values. Thus, instead of asking

“What is the probability that the light stays on for 9 seconds?”

one must ask something like

“What is the probability that the light stays on for at least 9 seconds?”

or equivalently

“What is the probability that the light stays on for a time between 9 and 10 seconds.”

(assuming T=10 s.) Since the interval from 9 s to 10 s makes up 1/10th of the total time span, the probability for getting an answer between 9 s and 10 s is 0.1. Similarly, the probability that the light stays of for a time between 1 s and 2 s is also 0.1.

10

Indeed, this is correct. my chart assumes that it’s between 1-x and x. So the “x=1” row is really “the light stays on anywhere from 0-1 seconds.”

11

I doubt there’s much chance of that.

Thank all y’all for the insights. If I can ask a final question, does anyone know of a site that would have good problems for someone interested in learning more about this? Especially like THIS ( more the odds and less the cards or the gambling.)