Why do odds of flipping a coin or rolling a dice change with progression?

easy weeks wrote:

Star wrote:

That's how gamblers think.

The roulette ball landed on red three times in a row so it will probably be black the next time.

But actual oddsmakers would think that way as well. If they had some sort of televised coin-flipping thing that you could bet on and the scenario I described above occurred, the odds would be made higher that it would heads a fifth time, simply because the coin turned up heads four times previously.

Oddsmakers don't set the odds based on the probability of something happening - though some might claim they do - they set the odds so that they get roughly the same number of bets on each side.That's why "the line" isn't fixed, but moves based on the action.

What about this? wrote:

How about applying this to a child being born? There are 2 possible outcomes - boy or girl (ignoring any transgender considerations). However, their are more females in the world. Isn't the probability of a girl higher?

Although some of the responses to this seem intelligent I don't think they answered your questions. I am also not sure if you are serious with the question or if you are trolling.

A binary outcome does not imply even likelihood of occurrence of the two possible outcomes. Just because at the moment of birth the child could be either a boy or a girl doesn't mean the likelihood is 50/50.

For example, if there is an implied birth rate of 52% female vs male, then at the moment of birth, you have a 52% chance of having a girl and a 48% chance of having a boy.

Going back to the coin flip example: if you have a weighted coin that produces heads more often then tails, each flip still has a binary outcome - heads or tails - but the likelihood of heads in this case is higher than 50%.

easy weeks wrote:

Star wrote:

That's how gamblers think.

The roulette ball landed on red three times in a row so it will probably be black the next time.

But actual oddsmakers would think that way as well. If they had some sort of televised coin-flipping thing that you could bet on and the scenario I described above occurred, the odds would be made higher that it would heads a fifth time, simply because the coin turned up heads four times previously.

To me you are now talking 2 different things.

I think that it is clear that with a "fair" coin the odds each time are 50/50.

But if you get a bookmaker to set the "line" then the bookie is really trying to get equal money on each side. You see this in a football game where one team opens up as a 10.5 point favorite. As money comes in on the underdog that line moves to maybe 8.5. Did anything about the teams change? (Let's assume no injuries or news?) No, but the oddsmaker needed to get money on the other side of the bet.

So yes, if in the TV coin toss the coin hit heads 3x in a row, a whole lotta people will be betting on tails and the oddsmaker will take into account how humans think in setting the odds.

Next up, the Monty Hall scenario...

luv2run wrote:

Next up, the Monty Hall scenario...

Always change your mind.....

Idiot wrote:

observer_of_things wrote:

Let's take this to a more extreme level. Let's say we've flipped it 100 times and it came up heads 100 time.

In finance, the difference between the Quant and the Trader can be summed up as follows: The Quant (aka the mathematician) insists the the probability that the 101st flip is still 50/50. The Trader knows better. He knows the game must be rigged and the coin is double-headed. He will bet anything that it comes up heads again. Generally, the trader is right about these things.

In your scenario, which is not based in math, as soon as the trader bets on a crooked flip the coin flipper changes to a double tails coin or doesn't take the bet. The trader doesn't control the flip.

Bayesian statistics has just as rigorous mathematical foundations as those for frequentist stats (whether or not either is really math is a different question). Working in a non idealized setting updating odds after iterations using Bayes' rule or some other method is perfectly reasonable and most likely more accurate than insisting trials fit some idealized distribution.

Idiot wrote:

observer_of_things wrote:

Let's take this to a more extreme level. Let's say we've flipped it 100 times and it came up heads 100 time.

In finance, the difference between the Quant and the Trader can be summed up as follows: The Quant (aka the mathematician) insists the the probability that the 101st flip is still 50/50. The Trader knows better. He knows the game must be rigged and the coin is double-headed. He will bet anything that it comes up heads again. Generally, the trader is right about these things.

In your scenario, which is not based in math, as soon as the trader bets on a crooked flip the coin flipper changes to a double tails coin or doesn't take the bet. The trader doesn't control the flip.

Bayesian statistics has just as rigorous mathematical foundations as those for frequentist stats (whether or not either is really math is a different question). Working in a non idealized setting updating odds after iterations using Bayes' rule or some other method is perfectly reasonable and most likely more accurate than insisting trials fit some idealized distribution.

Actually, since there is a chance it will not land on neither heads nor tails, it is not 50/50.

0%/10%

If you can answer this with an explanation you'll have your answer:

If the odds of winning a lottery jackpot is 1:100,000,000 and the cost of each ticket is $1 and you had $100,000,000 to spend on tickets, would you rather buy 100,000,000 "quick pick" tickets or would rather pick your own numbers and buy all 100,000,000 permutations?

Dice have no memory

What about birthdays? My friend and her sister have the same birthday and people always say "what are the odds of that"? I always respond 1/365. Am I right?

rf wrote:

Idiot wrote:

In your scenario, which is not based in math, as soon as the trader bets on a crooked flip the coin flipper changes to a double tails coin or doesn't take the bet. The trader doesn't control the flip.

Bayesian statistics has just as rigorous mathematical foundations as those for frequentist stats (whether or not either is really math is a different question). Working in a non idealized setting updating odds after iterations using Bayes' rule or some other method is perfectly reasonable and most likely more accurate than insisting trials fit some idealized distribution.

But this scenario the poster laid with 100 straight crooked flips has nothing to do with statistics. It's an environment where someone controls the outcome and decides what that outcome is based on a static input. Its 3 card Monty. An outside observer could bet on what the coin flipper will manipulate the outcome to be based on if it's a long term game or a one time flip, but the person betting on the flip doesn't realize he has imperfect information and the flipper has both perfect information and total control. It's a behavioral question.

The odds of a coin landing on edge according to wiki are 6000 to 1.

I vote 50-50 for the coin flip if random heights and rotations are used during flipping.

ehhhh wrote:

Although some of the responses to this seem intelligent I don't think they answered your questions. I am also not sure if you are serious with the question or if you are trolling.

A binary outcome does not imply even likelihood of occurrence of the two possible outcomes. Just because at the moment of birth the child could be either a boy or a girl doesn't mean the likelihood is 50/50.

For example, if there is an implied birth rate of 52% female vs male, then at the moment of birth, you have a 52% chance of having a girl and a 48% chance of having a boy.

Ehhhh, what you say is accurate mathematically, but it's not accurate demographically, and it's not the answer to the question what about this? asked. I answered his question.

The poster wanted to understand why, if there are more females in the world, the probability of producing a girl shouldn't be higher than .5. I answered his question using demography.

My answer is accurate. When you're asking a question about relative abundance in a population of organisms with life spans and differential mortality, then you cannot formulate an answer without considering when in the population life span you're looking for your answer. Note that I specified "cohort," which is required in order to make the problem tractable in this context, where we aren't professional demographers.

Your answer involved an imagined set of probabilities at the time of birth, probabilities that don't match what we actually find in the world among human beings.

Latsran wrote:

You're technically right, it's just that some mathematically illiterate people don't think that landing equal numbers of times on both heads and tails is equally as likely as all heads or all tails.

I'm a little surprised that no one has jumped on this statement, which reflects a different, but also common, misunderstanding of probability. Assuming a so-called "fair" coin, with 0.5 probability of heads and 0.5 probability of tails on each flip, the probability of all heads for n flips is 1/(2 to nth power), and the probability of either all heads or all tails is two times that number. For 100 flips, those are obviously extremely small numbers, on the order of 1/(10 to 30th power). The probability of equal numbers of heads and tails is very different, however, and the calculation generally involves the use of factorials. For 100 flips, the calculation is ((50 + 50)!)/(50! 50! 2^(50 + 50)), which equals 12611418068195524166851562157/158456325028528675187087900672, which is approximately .07959 (or approximately 1/12.56). So there's a pretty decent chance of 50 heads and 50 tails. The probability of equal numbers of heads and tails decreases as the number of flips increases (assuming, of course, an even number of flips, since the probability will always be zero for an odd number of flips); the probability of equal numbers of heads and tails for 1000 flips is approximately 0.025225, and the probability for one million flips is approximately 0.000797884, or approximately 1/1253.

But maybe that's just how "some mathematically illiterate people" think.

Since you all are interested in math I will repeat a question that was hashed out some time ago and let you all solve it since I don't feel like it.

What distances > 1 kilometer is it possible to run at an average sub 2:30 pace without running any actual sub 2:30 kilometers in the process?

For example, with 2000 meters it's impossible - if you run sub 5 then either the first half or the second half must be a sub 2:30.

But you can run 1500 meters as follows:

100: 13.0s

1300: 3:18.0 (198s), 2:32.3 pace

100: 13.0s

The fastest segments are 900/1300 x 198 + 13 = 150.07 = 2:30.07, not sub 2:30

but the overall time 3:44 is sub 2:30 pace.

Bookmakers make their money by adding in an overound so they offer odds as slightly less than actual probably of an event happening so over the long term they have a mathematical advantage.

Taking the coinflip example.. this is a 50/50 event so the true odds are evens or 2 (decimal odds), eg bet $1.00. If it wins you win $1 and get your stake back. What a bookie will do in this case is price the event as 1.9/1.9 for both both outcomes. So if the punter bets $1.00 and win they get their dollar back and win 90 cents. The bookie may have lost but they have paid out less than the true odds. Assuming they can balance their book (another punter betting $1.00 on the other outcome) they come out in front. They keep $1 and pay out 90 cents. This will be a 10% overround. Most books will operate much higher than this.. 20 or even 30% are not uncommon in big events.

easy weeks wrote:

Latsran wrote:

You're technically right, it's just that some mathematically illiterate people don't think that landing equal numbers of times on both heads and tails is equally as likely as all heads or all tails.

Right. And to take it further, it would be equally likely to have the same Powerball numbers come up 2x in a row than it would be for any other combination of numbers, right?

The likelihood of that same powerball number coming up a second time, assuming independence, is the same as the likelihood of any one other number coming up. It is not the same as any other combination.