Coin Toss: Not fair?

[Jamairoqui]

Has anyone studied the probability of the coin toss, and whether it is fair? For grins I kept track of 30 battles. Of those 30, I started first in 10 of them, and second in 20. There is only a 3% chance this could have happened if the coin toss is 50/50 (fair).

Can we accuse Rovio of “cheating” all of us by stacking the odds in the game’s favor so we have to spend LC to reset our birds after three losses?

Topic

[burbman]

I am not sure how “fair” the toss is, but it does tend to be extremely streaky. I have had runs where it seems I win the toss almost every time as well. I will track this over the next day or so and see how my numbers compare to yours.

[BF10]

I actually getting unbalanced tosses, and commonly I win the toss more than 50% of the time.

[Jamairoqui]

It is “streaky,” as a coin toss should be (meaning there is a 6% chance to get four heads or tails in a row), but I get streaks of six or seven “tails” in a row (there is less than a 2% chance of this happening), and streaks of maybe three “heads” in a row (I’ve never seen four heads in a row). This is exactly the sort of behavior that you see in realizations of binomial distributions if the probability of each outcome is not 0.5 each.

I suspect that the higher up the leaderboard you are in your league, the less fair the coin toss becomes. Which is fine if they want to run handicaps, but they should be up front about it.

[Jamairoqui]

@BF10: Have you actually kept track to see what your statistics are? That would be interesting to know. The sample size doesn’t have to be huge (although more is better), 30 battles is plenty.

[MVNLA2]

@jamairoqui — Seems to me you need a minimum of 100 tosses to draw any conclusions. If people keep rigorous track of all the tosses, including all those in non-ranked battles, we can combine the numbers. I would keep the non-ranked battles in a separate list, but if we’re going to keep track, I think it is important to track these also.
I’m in the same boat as @burbman in that there have been times when I seem to win every toss.
cc: @Dr_Ishmael @zackvoyager

[KillerKea]

I think the game slightly manipulates things to make leaderboards more competitive.

I get easy opponents more frequently when I’m not at the top of the leaderboard. Once I get to the number 1 or 2 spots I seem to get maxed out opponents a lot more often. Therefore it is harder to earn points when you are at the top which makes it easier for others to catch up since they are getting easier opponents. Of course I can get maxed out opponents when I’m in the middle or near the bottom of the leaderboard, but it certainly seems to occur less frequently for me. I haven’t kept stats. Does it seem that way to anyone else?

I’m guessing the game might pair people up based on their scores which would explain why opponents seem to get a little harder the higher your score gets.

[FreckledPig147]

I wish you could pick “Bird” or “Snout” at 50/50. Then, apparently we would have a much greater chance of going first, because I notice that I guess the coin flip quite accurately. (Like, after getting a small streak on either side , I expect the opposite to happen)

[KerrAvon]

Actually, @jamairoqui and @mvnla2, you would more likely need more than 1000 coin tosses to get meaningful results.

The probability of getting 10 wins out of 30 may be about 3%. But you have to put this into perspective. The chances of getting 15 heads results out of 30 coin tosses is less than 15%. In this example you have about a 43% chance of getting >15 wins and the same chance of getting <15 wins.

[burbman]

@killerkea – I have also noticed the opponent difficulty factor changing based on where I am on my own leaderboard. I made the assumption that the game was matching me up with players who were currently in the same star classification of their leaderboard and league as I was. I could be totally off base, but it made sense under the circumstances.

@jamairoqui – I am 20 battles in to my own tracking test, and I am not seeing anything out of the ordinary. 12 tosses lost, 8 won. Longest streak of three each way. I will keep tracking.

[Jamairoqui]

@KerrAvon, @burbman: The chance of getting any particular value is relatively small, but the chance of getting 15 +/- 2 heads is very large, almost 2/3rds of the time you will be within that range. The chance of getting very much less or very much greater than 15 is small. In most statistical analyses, significance is accepted at the 0.05 level. My test found 0.03 (a bit less even). It is highly significant. In fact, I ran 20 more battles, won the toss on five and lost on 15. This gives me a 15 to 35 ratio, and the probability of that occurring by chance is 0.2% (significance level of 0.002).

I don’t doubt that some people believe that have fair coin flips, because for small sample sets you can think the coin flip is fair (turn your example around a little, if the probability of heads is 1/3 and tails is 2/3, it is equally possible you would see 8 heads and 12 tails in that scenario as it is you would see 8 heads and 12 tails in a 50/50 game). But if you were at a casino, and red came up 35 times out of 50 on a roulette wheel, you could almost guarantee that the wheel was rigged.

To be mindlessly pedantic about this, 8 heads out of 12 rolls only happens 12% of the time for a fair coin. If you are betting on something with only a 12% chance of winning, you will go broke in a hurry.

[KerrAvon]

@jamairoqui I am sorry but those numbers are still not significant.

From your example of 30 coin flips, I think your number of approx 67% chance for the number of heads to be in the range 13-17 seems reasonable. But that still leaves 16% chance to get 12 or fewer heads and 16% chance to get 18 or greater heads. Those are still significant percentages. I wouldn’t be surprised in there was an 8% chance to get 10 or fewer heads.

You just had a bad day. I am sure if you try again in a couple of days you will get the complete opposite results. People tend to forget about the good days and just remember the bad days.

You really do need a lot more coin tosses to get a real feel for the percentages. Or perhaps a few players with lots (hundreds) of coin tosses. Completely independent results are always good. I am sure @burbman will provide some good numbers.

[Zack Voyager]

@mvnla2 @kerravon @jamairoqui @burbman Here calculations of the 95% chance frequency interval (you have a probability of 0.95 that the number of tails fall in this interval) for some numbers of coin tosses (I suppose that the probability is 0.5)

10 tosses -> 0.5±1/sqrt(10) -> [18.4% , 81.6%]
20 tosses -> 0.5±1/sqrt(20) -> [27.6% , 72.4%]
50 tosses -> 0.5±1/sqrt(50) -> [35.9% , 64.1%]
100 tosses -> 0.5±1/sqrt(100) -> [40.0% , 60.0%]
200 tosses -> 0.5±1/sqrt(200) -> [42.9% , 57.1%]
500 tosses -> 0.5±1/sqrt(500) -> [45.5% , 54.5%]
1000 tosses -> 0.5±1/sqrt(1000) -> [46.8% , 53.2%]

[MVNLA2]

@jamairoqui @kerravon @zackvoyager — Actually, I do agree with KerrAvon, in that even at 1000 tosses the range of the 95% interval is quite large. We saw this when tracking statistics for the Dojo offerings. I picked 100 tosses, because I thought you might get some idea with that number. Not surprisingly 40 to 60 percent is a larger range than I expected.
I too have started tracking coin tosses, and so far, even with a small number (~30), it is very close to 50/50.
The spreadsheet I’ve started has the following columns:
League / Date / Position in league / Ranked Battles You win / Ranked Battles Opponents Win / Banner Strength / Bird Rank / Unranked Battles you win / Unranked Battles Won by Opponents
I separated tosses from ranked and unranked battles in case they turn out to be different. I am tracking banner health and rank of birds to see if there is any correlation with position in league.

@Jamairoqui — What league are you currently in (wood or stone)? I am still in wood, but should reach stone at the end of the week.

@KillerKea — As I said I’m still in wood, but all the data I’ve collected so far was taken while I was #2 in my league. The opponents banners have ranged from 3342(!) to 11200 (mine is normally 11139, and my birds are all rank 10). The opponents themselves have ranged from one bird at rank 10 or 3 birds at ranks 2,2,1 to 3 at rank 10. Haven’t calculated an average yet.

[Immortal… Larry]

It’s mostly a matter of perception. You tend to *forget* the winning tosses simply because the battle is finished faster and there’s nothing out of the ordinary. Each lost toss tends to be remembered since you have to work harder towards your goal.

A while ago I recorded more than 900 tosses and came out at 49,8% losses. I don’t have the sheet anymore, but it covered the leagues from gold to diamond for many weeks.

[BF10]

@freckledpig147 – But you can always pick the bird and you can win almost anything. So that can make opponents cheat to pick the bird every time and win.

[burbman]

Well, the game locked me up for the third time today. So my recorded totals, (I played some additional battles but was away from my notepad) came in as follows:

Win the toss 34
lose the toss 41

Longest streak – 6 wins.

Conclusion – based on this, and @Larry’s observations the coin toss is fair, and sometimes the player is just unlucky.

[MVNLA2]

@larry — Thanks, I had forgotten your data. Guess we don’t need more statistics.

[Immortal… Larry]

@mvnla2, one person’t record should never go unchecked, so feel free to start a new statistic on this… :)

I just want to emphasize the idea that people mostly start getting frustrated about *unfair* tosses after getting a long streak of losses. People often just forget about the long streaks of wins the day before.

[MVNLA2]

@larry — I stopped tracking toss statistics as soon as I was reminded that you did so. Not clear how much longer I’m going to play the arena. Maybe I need to wait a week or two for all the diamond-leagers to clear out of stone league. I seem to spend as much time getting new opponents as I do fighting them.

[WhiteAsIce]

I’ve been tracking my coin tosses since v1.2.3. I’m currently at 17% I start, 83% enemy start.

[burbman]

v. 1.2.3???

[WhiteAsIce]

Yes, since v1.2.3

[KerrAvon]

@whiteasice Are you playing regularly? Or is this just a few battles? If you have played more than a hundred arena battles, then you definitely have a problem with your coin toss.

[WhiteAsIce]

Just played 15 matches today to complete the dailies. Lost all 15 coin tosses.

I think the coin toss is a stupid idea, and Rovio should get rid of it. We all know that we’re not actually playing other people – we’re playing against the AI using teams uploaded by other people.

@kerravon I play at least 5 matches everyday to complete the dailies. I log my coin tosses every single match. After today’s dailies, I’m at 78 coin toss wins out of 465 total matches, so in 16.77% of all my played matches, I attacked first.

[KillerKea]

I think there is a bug either in some types of devices or some versions of the game where the coin toss is not fair. I remember early on a lot of people talked about losing most of their tosses and this was true for me too, but they fixed that bug and it has been roughly even for me for a long time now.