Let me know if my numbers are off...

But just going with these numbers for argument sake, let's say you're watching an NBA game. The combined score after Q1 is 'odd.' There is a 1/8 chance of Q2-Q4 all being 'odd' also. If it isn't the case it's a hit and if all the rest are 'odd' it's a miss.

is it fair to treat this as a binomial distribution - not so much "is the score going to be odd or even" because I already figure that can be calculated as a B.D. but, - whether the game is a miss or hit? A miss has a 12.5% chance.

Can I use the calculator at https://stattrek.com/online-calculator/binomial.aspx
trials being a game, 0.125 as the probability, success being a 'miss?'

If so I'm getting
Games Probabilty
1 12.5%
2 21.9%
3 28.7%
4 33.4% etc

...when I hit 15 games, the Probability begins decreasing. Am I considering the wrong probability box? Would it be more accurate to just do P(miss)G1 or P(miss)G2 or...and so on?

I’m not sure what you consider to be a miss (= success). It seems like it is Q2-Q4 all odd given Q1 is odd. You gave that a (1/2)^3 = 12.5% probability. Then you correctly used the binomial to find the probability of exactly one hit in N games.

If the question is, for example, what is the probability of a hit in G1 or a hit in G2 or a hit in both, that is a different calculation, e.g., 2*0.125 – 0.125^2 = 0.2344 = 1 - 0.875^2 if both games meet the Q1 condition, whatever that is.

Issues:
1. The above assumes if the Q1 score is even then that game is not considered.

2. Given Q1 is odd, is the probability each subsequent quarter has a 50% probability of being odd. I would strongly think so but can’t be 100% certain.

3. Even more unlikely is the possibility that the games are not independent or the probability is not constant, two conditions needed for the binomial to apply.

I hope somewhere in this response I answered your question.

Thanks
You could call it either way but for discussion any Q2-Q4 that is the same as Q1 (odd or even) is a miss. If any quarter Q2-Q4 is opposite of Q1 then that's a hit. In this case a hit is a win because I am betting against all quarters being the same. Miss is just in relation to a team hitting a different O/E value from Q1.

I don't know if this is an ideal strategy but I typically see what happens in Q1. If it's odd I bet against odd hitting in all of the remaining qtrs. If Q1 is even, I bet odd until odd hits for the rest of the game.

Looks like you found the "Cumulative probability:
P(X > x)" for 2 games according to the calculator, but I am considering odd or even in Q1 and a hit being the opposite hitting any time, Q2, Q3 or Q4. It only has to hit in one of those.
How does that change things? I'm guessing it will be one of the other values on the calculator, probalby 0.219

so yeah,

1) even or odd are in play depending on Q1.
2) Yes it is hard to say for sure if the probability is 50% I feel it is very close if not. But all things considered seems as if one quarter to another is roughly 50 or at least independent. One thing I am considering is that 1 team may have a higher tendency to score odd or even than another and this of course would mean that one matchup in a game may be further from being 50-50 than another. It just feels like 2 particular teams set the stage with whatever happens in Q1 and then they play it out from there, but say like if Q2 misses, then Q3 misses and then Q4 misses, whether it is the case or not, it feels more in line mathematically to let the next game's Q1 play out and to go from there instead of playing straight into Q1 from another game's Q4. I may be wrong with that but it's just the feeling.

I am compiling stats by team on how often the team hits, in which Q and how often they miss. So far Q2-Q4 and miss numbers seem to follow expected probabilities (Q2 roughly 50%, Q3 roughly 25%, Q4 roughly 12.5%, and miss altogether 12.5%).

I was just going over the LA Lakers games and they seem to be an outlier. So far 1 miss in 36 games with still about 9 games to look over. Otherwise, so far, stats seems to be in agreement by team.

Eventually I would like to figure out how to calculate regression to see specifically how hot or cold a team is running on the metric. So far I fade a team that has gone over the average amount of games to get a miss and expect a team high on misses to get some hits. I'm in the initial stages of getting that data and applying it. So far, probably have stats on 10 or so teams. And so far no one has missed 3 games in a row. Had a couple miss 2 in a row. Knicks played last night, they had missed the previous 2 games so I had them as a lock of sorts to hit, but also don't have much faith in those numbers yet. They hit in the 4th Q against a Pelicans teams that had gone about 8 or more games without a miss. So yeah, hoping to get more stats and numbers to help

3. You're thinking it's likely that the games are independent and that the BD applies? Let's assume it does and see what we get.


[Update on LAL they had 2 misses in a row in the first 6 games of the year and then have had 1 miss in the last 40]

[Also using the binomial N=1 (game) P (X=x)= 0.125 that's the probability of a miss right...which i guess obviously makes sense if I put that the probability is 0.125...wondering why the probability decreases around ~15 games. I'm guessing maybe that is the probability of only having 1 success (or miss)]

You modeled each quarter as being 50%, which implies that Q1 has no influence. But if you're saying Q1 does have an influence, then you need to look at historical data to quantify it, since Q2-Q4 aren't Binomial(p=.5) if your hypothesis holds. However, if you only believe it about particular teams, you might run into the problem of a small sample size.

It boils down to whether the teams will make an even or odd combined number of threes and free throws. The only way the even/odd probability isn't 50% is if time is a factor. However, I'd be surprised if 12 minutes isn't enough time to make the probability extremely close to 50% regardless of the team matchups.

That's Gambler's Fallacy, unless books are stupidly adjusting their prices to the point where you're getting a good price. Each game is independent.

You calculated the chance of exactly one success, which begins to decrease after n=8. Statman showed how you'd calculate the chance of at least one success. However, I'm not sure why you'd wanna know either of those. It sounds like you're more interested in the average, which is simply N/8, though I wouldn't call that useful info either.

Thanks.
I just mean teams set the stage with Q1 by whatever they score - odd or even.
Definitely trying to get more data anyway, at least on the current season.

Prices for Odd/Even are ALWAYS -110/-110, starting out. They aren't adjusted for anything else other than if a team already has Odd or Even and you are putting in a live bet (can't bet quarter totals until the quarter starts. When a team has the value you are choosing the line might become -115 vs -105.). As far as Gambler's Fallacy I definitely see it when the game gets to Q4 and Q1-3 are OOO or EEE, the line is always adjusted to like -120 or -125, for the value that has yet to hit. Typically one can get the Q1-3 value for Q4 for +100. Looks to me by all accounts that at that point it's 50-50 no matter what has happened so far in the previous quarters and they are just encouraging ppl to take a side for a bad price because IT HAS TO HIT NOW.

I can see where you are coming from if something is 50-50 it is going to be 50-50 each trial no? But can we not use longterm stats to see trends and times when regression is due? In the end, do probabilities not return to their average? I see both sides so it's confusing to me. If something is running outside of the norm does it not fall back into standard deviation? That's what I'm trying to figure out.

If a team is running a stat at 6.25% that should be hitting 12.5% is it not more likely that they are going to hit that value. 40 games in a row missing a 12.5% chance...it's hard not to think that the chance is due. I will actually be happy if that is not the case.

I think I'm just wanting to have my math checked and see if there is any other way of looking at it say like if the Binomial calculator can help.
Interested in figuring out more about regression, standard deviation etc and how it would apply
I am using the martingale betting strategy so I think my main concern is what are some odds on how many times a hit/miss is going to occur in a row and what is the likelihood of each number.

I like that.

Consider flipping a fair coin. The first five flips are heads so using that data alone, you would say the probability of a head is 100%. The next 5 flips are 3 heads and 2 tails. Now P(H)= 8/10 =80%. The next 5 flips show 1 head, 4 tails so P(H)= 9/15 getting closer and closer to 50%.

In other words small samples can show “Odd” results but if the probability stays constant (in this case 1/2), the coin has no memory and the sample mean will approach the theoretical mean as the sample size increases. Your phrase “is due” is exactly what the Gamblers Fallacy refers to.

If in 40 trials you observed 6.25% when 12.5% was expected, that means 2 or 3 hits were observed. Using the binomial calculator, the probability that 3 or less hits will occur in 40 trials is about 25% so that is not so unusual. Now if the same result occurs on 400 trials, the probability is about 1 in 1000 so you would have to suspect the 12.5% calculation or bias of some kind in experiment

I meant to write 100 trials.

I get that. Saying something is due is pretty similar to saying that the sample mean will approach the theoretical mean as the sample size increases, no? How else is it going to approach the mean over the course of more trials without the data being heavy on the opposite side? I know what you mean though, sample size is crucial. In the NBA each team plays 182 games a season which is actually a lot, I believe, compared to a lot of other sports. Not an astronomical sample size.

Binom. Distrubution leads into statistical theories like standard deviation, regression, etc, no? It's been a while since I've been over any of that. I like the example you gave though. So let's say 48 games, N=48 right? Successes, yes, =3. Here's the thing though. For all 4 Qtrs to hit the same value, O or E, that is .5^4, am I right? or 1/16. Which actually is 6.25%. The only time Q2-Q4 miss all 4Qs miss also, but for my strategy I have only been concerned with Q2-Q4 or a 12.5% chance. So maybe that is normal after all and the other teams that are hitting around 12.5% are actually running a little hot but definitely not significantly over the sample. But yes, for the 12.5% to apply I would also need to include games where Q1 was different from Q2, I believe, and then from there when Q2-Q4 were all the same. Sorry for the confusion, I'm a little tired and I don't think I have explained this all very clearly.....

Regarding the calculator, to make sure I'm using it correctly (am I correct to think now that it is mainly used to check numbers vs predicting numbers?)....
I put in s=.0625, N=48, x=3 , and get 0.23136260559 which I believe is what was shared earlier. Guess I would use a binomial table to check how much this was in line with the theoretical mean, but obviously that is a good number.
How far off exactly is 0.125, N=48, x=3 P(x)=0.08299 out of curiousity?

How can I figure out how many Quarters I can expect to miss (be the same value, odd or even) in a row and at what probability?
What about sets of 3-in-a-row?

You flip a coin 10 times and it's heads every time. So far 100% heads. Suppose we flip another thousand times and the lead of +10 heads stays the same. That is, our next thousand flips are perfectly split 500-500 in alternating fashion, making the score 510-500. Check it out: 500/1010 = 49.5%, so tails went from 0% to 49.5% without the coin compensating on the tails side at all. Now do another 10,000 flips with another perfect split. That puts tails at 5500/11010 = 49.955%. You get the picture: tails isn't due, but regression to the mean occurs anyway.

If it's a streak of k in a row misses, the chance of it happening in the next k trials is q^k where q is the probability of a miss. If you want the chance in the next n trials where n>k, there isn't a human-friendly formula, nor do I see a good online calculator for it (eg the SBR one is wrong). I have my own code to calculate this if you need me to plug in some numbers for you.

Avg length of miss streak = P(exact length 1) + 2*P(exact length 2) + 3*P(exact length 3) + ...
If the streak has to start on the next trial: q(1-q) + 2(1-q)q^2 + 3(1-q)q^3 + ...

When talking about things like this, I always feel it's important to point out that absolute difference between H/T will probably always grow while the % will still approach 50%. If your 1,000 flips came up H 510 and T 490 and the 10,000 flips 5,010 and 4,990 we'd still be regressing to the mean.

True good point. It will most likely grow, and the keyword you mentioned is "absolute" because it can almost as easily be Tails with the >10 lead after another large batch of flips.

Here’s a way of looking at regression to the mean with a continuous (non-binomial) variable.

Suppose with a small population, say a company of 100 employees, a survey of all employees accurately reports an average weight of 150 pounds. Someone questions that and starts his own survey of all employees to be done the same day. Of the first 10 people weighed, the average weight was accurately reported as 200 pounds (He started in the company warehouse.) Clearly as the doubter continues weighing people the average will regress to the true mean.

The extreme early measurements are eventually balanced by the less extreme or opposite extreme ones. The error in thinking is that the extreme event caused the less extreme one – the 200 lb warehouse guy caused the secretary to the VP to only weigh 110 lbs.

Clearly, if the average of a variable in a population is X, then the measured average of all in the population will be X. Of course in practice, you rarely know the true population value but as you randomly sample more, the sample mean will approach the true value if the influencing conditions don’t change,