Sequenced Events and Statistical Bias?

Factual Questions
1

I was wondering how do we know if the odds of a subsequent event INCREASE, DECREASE, or perhaps even remain the SAME after an initial event?

Sometimes, we look at an event and say the odds of a repeat event are now even more likely (increased odds). Other times, we say the odds of a repeat event are now much less (decreased odds). And yet, couldn’t the odds of a subsequent event possibly be the same as the first event?

Are we employing a little bias into the calculation of stats? While I WAG that an increase in odds suggests some relation between events, and a decrease in odds might imply no relation…yet isn’t also possible for the odds to remain the same?

For example, you might say the odds against a second “hundred years” floods occuring within the given period is much greater than the odds against the first. But, if I were to hit the lottery once and then hope to hit again that year, wouldn’t each try have an equal chance?

Maybe I’m mixing up some facts. I hope someone can explain! :confused:

“They’re coming to take me away ha-ha, ho-ho, hee-hee, to the funny farm where life is beautiful all the time… :)” - Napoleon IV

2

For events which are causally independent of each other (like flips of a fair coin), the probability of future events does not change based on past events.

For events which are causally dependent on prior events (e.g. earthquakes), the probability of future events can change based on past events. By definition, past events cause a change in future events.

For events in which it’s unknown whether or not they’re causally related, a statistically unlikely run of events can suggest a causal relationship. However, to prove a causal relationship, you must hypothesize a cause, make a statistically unlikely predition based on that cause, and see that predition borne out by actual events.

Catapultam habeo. Nisi pecuniam omnem mihi dabis, ad caput tuum saxum immane mittam.

3

…make a statistically unlikely prediction…

4

This is like the “theory” of hot/cold numbers in lotteries.

At the local drug store, they post a list of “cold” numbers for Pick 3 and Pick 4 Virginia lottos. The “cold” numbers that haven’t appeared yet has just been happenstance. Any given drawing has a equal chance of drawing a particular number.

There have been not near enough trials (drawings) for the Pick 3 and Pick 4’s to indicate a trend in the drawings. Even after a century, each Pick 3 combo will statistically come up 36-some times. If a given one only comes up 18 times in the next century, that’s still not significant.

You must unlearn what you have learned. – Yoda

5

Yeah, the distinction between independent and dependent results is profound and not always easy to make. The example I heard from one of the SDMB mods was, say somebody told you there was a fair coin, and they flipped it 99 times, and it came up heads each time. What is your guess as to the result of the hundredth toss?

Your guess should probably be: They’re lying. It isn’t a fair coin. The hundredth toss will be just as bogus as the other 99. But that’s a practical, not a statistical judgement. I mean, while math tells us the odds of 99 flips coming up all head (1 to 2^99 against, IIRC), it’s only our judgement of the overall situation which leads us to believe the coin is rigged.

Most real-world probability cases are not perfectly independent, but a lot of them are so close to independent that it doesn’t matter. Let’s say you’re counting automobile makes at the center of town. Fifty Mazdas cross the bridge going south before noon (none of them come back). Is this going to affect the odds of a given number of Mazdas crossing the bridge going south after noon? Yes, because a limited number of Mazdas have been manufactured. This, however, is probably a negligible effect, and can be ignored.

The other effect, way too big to be negligible, is on the number of Mazdas going north in the afternoon. You don’t expect a “net southward Mazda migration”, after all, you expect people to go home after work.

The point is, it takes a lot of thinking to figure out whether or not a given sample is going to be close enough to independent. Coin tosses (with fair coins) are a good example of total independence; counting the number of women in a town square is nearly independent; counting the number of adult women in a nuclear family isn’t even close to independent (having already counted Mom drastically reduces your chances of counting another mom in a single-family household).

Hopefully, I can convince you to accept “hopefully” as a disjunct adverb.
Frankly, I would be lying if I said I were confident.
Perhaps this subject is simply too complex for me to explain.
Unfortunately, I would be lucky to explain my way out of a paper bag.