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

0/10

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.

It is 50/50 every time.

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.

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.

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?

Yep.

You need to understand dependent and independent probabilities.

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?

Yes, assuming powerball number selection is truly random.

Separate coin flips, unweighted dice rolls, true random number generations, etc. are independent events. The previous result has no impact on the current trial.

If you flip a coin 99 times and it lands heads each time, the chance of it landing heads the 100th are still 1/2.

The cumulative chance of 100 independent events with 50/50 odds all giving the same result is (0.5)^100, but any single event trial has exactly the same odds as the previous one.

Thanks for the explanation. I knew this instinctively (seems self-evident and obvious to me), but couldn't articulate it very well. Was trying to argue it to a friend who doesn't understand the concept of independent events having no bearing on the next one.

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?

Wrong, and that's the aspect of this that people don't understand when they spread this distorted reductive line of thinking. It's a big example in a lot of undergrad-level "Statistics 101" type courses, even now, because it sounds flashy for "statistics" and brings classroom discussion.

The nuance is that "It's just as likely for the next (1) number drawn to be the same as the last one was, as any other individual number". This may sound the same as "It's just as likely for it to be two (2) in a row of the same as any other", but it's a matter of context. There are a lot of flashy "logic" problems like this where it's really a matter of verbal language being imprecise (another famous one being "Is the infinite decimal .9999... the same as 1").

it's not econometrics wrote:

easy weeks wrote:

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?

Yes, assuming powerball number selection is truly random.

Separate coin flips, unweighted dice rolls, true random number generations, etc. are independent events. The previous result has no impact on the current trial.

If you flip a coin 99 times and it lands heads each time, the chance of it landing heads the 100th are still 1/2.

The cumulative chance of 100 independent events with 50/50 odds all giving the same result is (0.5)^100, but any single event trial has exactly the same odds as the previous one.

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?

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?

Consecutive power ball picks are not independent: each pick does change the odds of the next pick since you are removing a ball from the pool.

And in the coin flip scenario book makers know the flip is 50/50 but the odds would not simply reflect that since betting lines are set to attract roughly equal money on both sides. If someone bet $1m on heads the casino would change odds to attract money on tails.

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?

Saying there are more females is not a comment on birth rates.

You can't say "there are more females in the world" unless you specify at what age(s).

If you have a cohort of newborns, then there will be more males than females. Thus the probability that a given newborn is male is higher than the probability that it's female.

If you have a cohort of old people, then there will be more females than males. Thus the probability that a given old person is female is higher than the probability that a given old person is male.

The reason is that at every age the probability that a male reaches the next age is smaller than that a female reaches the next age. This is typically called "differential mortality by sex."

In this way, the proportion of males (over time, always over time) is lowered and thus the proportion of females is increased.

As in so many problems, you can't make sense of it unless you learn to think about it over time, not as a snapshot.

On some level OP is right. If someone were to flip a coin 100 times and hit heads each time then the odds of there being some form of bias interfering with the toss are much more likely than the 1/2^99 chance that they flipped the same outcome 100 times in a row. Maybe the person subconsciously favours starting with the tail side up and flips the coin 11.5 rotations almost every time or something. Basically after a certain amount of tosses if you see results that are too many standard devs from the mean then you can assert that the true odds are not 1/2 for each face. You would be a fool not to favour the more common result.

The more you flip the coin or roll the dice the more probable one of the outcomes is. That's because the coin/dice gets worn out and will tend to land more on one side than the other(s).

easy weeks wrote:

What I mean is this - if you flip a coin once, we can assume about a 50/50 chance of either side coming up.

But if you flip it say, five times in a row and heads comes up the first four times, the odds will be more in favor of tails coming up the fifth time, right? And that's what I don't understand. The coin doesn't have a memory. It doesn't know that it turned up heads the last four times. To me, it's 50/50 every instance you flip a coin, no matter how many times you flip it. Why do I see a conflict and what am I missing here?

Actually there is a miniscule higher percentage that it will come up heads again.

easy weeks wrote:

What I mean is this - if you flip a coin once, we can assume about a 50/50 chance of either side coming up.

But if you flip it say, five times in a row and heads comes up the first four times, the odds will be more in favor of tails coming up the fifth time, right? And that's what I don't understand. The coin doesn't have a memory. It doesn't know that it turned up heads the last four times. To me, it's 50/50 every instance you flip a coin, no matter how many times you flip it. Why do I see a conflict and what am I missing here?

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.