Probability versus Odds: Revisited?

Question

Each day, there are a number of events that can happen. The MOST PROBABLE event each day has a "probability" of less than 50%. Therefore, the ODDS of the MOST PROBABLE event happening are LESS than the ODDS of the most probable event NOT happening. However, the data shows that the MOST PROBABLE event each day actually occurs 68% of the time. Statistically, what can be said of these event outcomes each day, and what is the best way to determine whether the MOST PROBABLE event will or will not occur on any given day?

This is not a trick or exotic question. What I'm basically saying is that the MOST PROBABLE event, even when it's probability is less than 50% each day, actually occurs 68% of the time. I'm wondering what, if anything, this suggests about my data. In other words, if you had a series of poker matches over 5000 days, and the FAVORITE in each match had a probability of winning that was less than 50%, what phenomenon might explain why the FAVORITE is winning 68% of the time over those 5000 days?

Answer 1

If the data over 5000 trials is showing a 50% probability event occurs 68% of the time, I'd say there's likely to be an error in determining the probability at 50%.

Consider a binomial distribution. The number of trials is n, the probability of an event happening is p, the probability of it not happening is q.
The mean number of times the event should happen is m, where
m = np. [In your case, it's 5000 × 0.5 = 2500.]
The standard deviation of occurrences is s, where
s = √(npq). [Again, for you, this is √(5000 × 0.5 × 0.5) = √1250 = 25√2, or approx. 35.355339.]
For 68% (or 3400) of your trials to succeed, this is euqivalent to a z-score of
z = (x - m) / s = (3400 - 2500) / (25√2) = 900 / (25√2) = 18√2, or approx. 25.456.
This is to say it's 25.456 standard deviations above the mean. This would be the statistical equivalent of someone having an IQ of 482 (which has never happened) or a woman being 10'8" tall (which has also never happened).

I'd say your estimate of 50% probability is not figured out correctly. It should be a lot closer to 68%, based on your observation.

Answer 2

To respond to your revised question:
You're still misunderstanding the language of statistics and probability. Probability is the number of times an event occurs out of the number of opportunities. The key source of confusion seems to be in how you define what an opportunity is.

If you define probabilty the same way for both numbers you presented, then your problem is that your making incorrect calculations. Otherwise... it is not clear in your example whether either number (50 or 68) is the 1) number of days an event occurs / X days (e.g., 365 for a year), 2) the number of times the event occurs during one day / the number of "times" (e.g., minutes) in a day, or 3) some other definition.
Until you can sort out what your terms mean instead of just writing "probable," we can't really understand or answer your question, and my guess is that you haven't thought about that enough. Good luck.

Answer 3

hmmmm... probability of a flipped coin landing with heads up is 50% (assuming a balanced coin)... flip it one time...

if it comes up heads... you have 100% of the time heads
if it comes up tails... you have 0% of the time heads...

the 50% would not be reached until you do a VERY LARGE sampling of data.... and.. if you did collect a very large amount of data for the coin toss and it came up to 55%... your method of flipping is flawed, or coin unbalanced.

same with your example... either you do not have enough data collected yet.. or... there is a flaw either in the method or in the given probability.

Answer 4

The events can not be independant. Or the cards are marked and the best player is the only one to know it.

Answer 5

You gota get up to get down!