# Law of Averages and Coin Tosses **URL:** \ **Category:** Factual Questions\ **Created:** [September 27, 2001, 2:31am UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046 "2001-09-27T02:31:08Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![zut](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zut/32/2875_2.png) [@zut](https://boards.straightdope.com/u/zut)\ **Post date:** [September 27, 2001, 2:49pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/21 "2001-09-27T14:49:25Z")
> [@](#): > > \*Originally posted by Another Primate \* > How about this – I’ll ask her which ones she **would** choose, and you can ask the lottery for their reciprocals. Don’t take “no” for an answer. Or, how about asking her to choose 45 numbers from 1 to 50, and 35 numbers from 1 to 36? If you tell me which ones she _doesn’t_ pick, I’m willing to perform a simple experiment to determine your daughter’s anti-precognitive abilities. ---
**Author:** ![Another\_Primate](https://avatars.discourse-cdn.com/v4/letter/a/f19dbf/32.png) [@Another\_Primate](https://boards.straightdope.com/u/Another_Primate)\ **Post date:** [September 27, 2001, 2:53pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/22 "2001-09-27T14:53:55Z")
Dear zut, I’m humbled. (Again.) Good idea. Your wish is my command. Sue ---
**Author:** ![John\_Corrado](https://avatars.discourse-cdn.com/v4/letter/j/a88e4f/32.png) [@John\_Corrado](https://boards.straightdope.com/u/John_Corrado)\ **Post date:** [September 27, 2001, 3:19pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/23 "2001-09-27T15:19:09Z")
> [@](#): > > \*Originally posted by Afghan \* > \*\*personally I think , that there is really really no such thing as 60% chance that this will happen or 40 % chance that this will not happen. All probablities are just 50% , either you will pick a red ball out of 100 blue balls (without seeing ) or won’t. Why so much fuss about probablity \*\* Either the sun will come up tomorrow, or it won’t. 50/50 chance either way. ---
**Author:** ![panamajack](https://avatars.discourse-cdn.com/v4/letter/p/47e85d/32.png) [@panamajack](https://boards.straightdope.com/u/panamajack)\ **Post date:** [September 27, 2001, 6:26pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/24 "2001-09-27T18:26:22Z")
The comments so far have only barely skirted on the philosophical issue of what probability actually is, which might has some bearing on the question (and is possibly the source of **Afghan** ’s problems). I’m not intentionally trying to muddy the waters here – nothing I’m saying contradicts what’s been said (the 50/50 thing aside). This is just a slight diversion into the epistemological problems of probability. What exactly does it mean for you to say, “The coin has a .5 probability of coming up heads”? Suppose you created a ‘perfectly balanced coin’. How would you know this? Can you really claim any advance knowledge of random events? You could say that this number comes about from long-term statistics – in other words, in various 10000-flip trials, a perfect coin comes up heads at an average of 5000 times. But if we say that the probability is 50%, then any _particular sequence_ of heads is equally likely to occur in any particular trial. We might get a result like Casey Primate. One short trial is not enough, but how many trials (and how long) is enough? This approach leads to a question when considering single events. Looking at **Afghan** ’s example, you have the bag with 100 blue balls and 1 red ball. You pick exactly one ball, then never pick again. How do you determine the probability of picking the red ball? If you say 1/101, or even 50/50, how do you determine this? The philosophical definition of probability is not satisfactorily solved – at least not in my opinion. You may choose whatever theory you wish; the mathematical side is the same. ---
**Author:** ![panamajack](https://avatars.discourse-cdn.com/v4/letter/p/47e85d/32.png) [@panamajack](https://boards.straightdope.com/u/panamajack)\ **Post date:** [September 27, 2001, 6:32pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/25 "2001-09-27T18:32:59Z")
corrections : I said, “if we say that the probability is 50% …” This is true no matter what the probability. It is essentially a fundamental assumption that any particular sequence of events is equally likely. But it is this that perhaps leads to the problem of what probability is. Also, I pronounce ‘has’ with a hard ‘s’, something like ‘v’. ---
**Author:** ![Wumpus](https://avatars.discourse-cdn.com/v4/letter/w/8c91f0/32.png) [@Wumpus](https://boards.straightdope.com/u/Wumpus)\ **Post date:** [September 27, 2001, 6:52pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/26 "2001-09-27T18:52:04Z")
While no one would claim it solves all philosophical problems with probablity, Bayesian probability theory was created specifically to deal with the how-can-I-tell-if-it’s-a-fair-coin scenario. Each time you flip the coin, you update your beliefs about the probability of its coming up heads: > **[Centre for Software Reliability | City, University of London](https://www.city.ac.uk/research/centres/software-reliability)** > > The activities of this centre mainly concern the assessment and assurance of the dependability of socio-technical systems. ---
**Author:** ![Another\_Primate](https://avatars.discourse-cdn.com/v4/letter/a/f19dbf/32.png) [@Another\_Primate](https://boards.straightdope.com/u/Another_Primate)\ **Post date:** [September 30, 2001, 4:36am UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/27 "2001-09-30T04:36:13Z")
> [@](#): > > \*Originally posted by zut \* > \*\*Or, how about asking her to choose 45 numbers from 1 to 50, and 35 numbers from 1 to 36? If you tell me which ones she _doesn’t_ pick, I’m willing to perform a simple experiment to determine your daughter’s anti-precognitive abilities. \*\* **zut** , here’s Casey’s reply: "Make sure he definitely doesn’t pick 17, 38 or 47 since I tried to guess those about 4 times each. "2.10.18.28.30. (37,39,43 are alternates since I only got those after trying and TRYING to find numbers that I hadn’t guessed yet.) "will do the 36 thing later. “casey” ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 30, 2001, 8:44pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/28 "2001-09-30T20:44:03Z")
OK, now I’m waiting for the lottery to come up 1,11,17,38,47. Either that, or 2,10,18,28,30 will win in Casey’s state, but not in **zut** ’s. ---
**Author:** ![Another\_Primate](https://avatars.discourse-cdn.com/v4/letter/a/f19dbf/32.png) [@Another\_Primate](https://boards.straightdope.com/u/Another_Primate)\ **Post date:** [September 30, 2001, 9:44pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/29 "2001-09-30T21:44:04Z")
> [@](#): > > \*Originally posted by Chronos \* > \*\*OK, now I’m waiting for the lottery to come up 1,11,17,38,47. Either that, or 2,10,18,28,30 will win in Casey’s state, but not in **zut** ’s. \*\* Well, Casey’s from Oregon, and our lottery (they call it Megabucks) doesn’t work like that. How _does_ ours work? I’m not sure… a few times a year, I’ll buy a $1 quick pick, throw it in my glovebox, and forget to ever look at it again. She thinks **zut’s** lottery must have some kind of ‘powerball’ thing, and wants to know, before she does her “pick 35 numbers from 1 to 36” thing, whether the number she ends up not picking can be 2, 10, 18, 28, 30. She thinks so, because otherwise **zut** would have said so. Ramble, ramble, ramble… OK, let me say this differently. Do the balls all come out of one bucket? Or two? Sue ---
**Author:** ![zut](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zut/32/2875_2.png) [@zut](https://boards.straightdope.com/u/zut)\ **Post date:** [September 30, 2001, 10:36pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/30 "2001-09-30T22:36:10Z")
As far as I know (I don’t play the Lottery more than once a year), the 1-36 thing is independent of the five 1-45 numbers. The [MI Lottery Commission’s Big Game page](http://www.state.mi.us/milottery/games/OG_biggame.htm) seems to confirm so. I anxiously await the final number that Casey does NOT pick. ---
**Author:** ![Doctor\_Goo\_Fee](https://avatars.discourse-cdn.com/v4/letter/d/3d9bf3/32.png) [@Doctor\_Goo\_Fee](https://boards.straightdope.com/u/Doctor_Goo_Fee)\ **Post date:** [September 30, 2001, 11:06pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/31 "2001-09-30T23:06:27Z")
Bell Curve. That would help to answer some of these probability questions. I’d explain it myself 'cept my memory of college stats class is f u z z y. ---
**Author:** ![Another\_Primate](https://avatars.discourse-cdn.com/v4/letter/a/f19dbf/32.png) [@Another\_Primate](https://boards.straightdope.com/u/Another_Primate)\ **Post date:** [October 1, 2001, 12:12am UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/32 "2001-10-01T00:12:19Z")
> [@](#): > > \*Originally posted by zut \* > \*\*As far as I know (I don’t play the Lottery more than once a year), the 1-36 thing is independent of the five 1-45 numbers. > I anxiously await the final number that Casey does NOT pick. \*\* Drumroll, please… Here’s the answer, straight from the anti-precognitor’s electronic mail reply: “Powerball # =18 (or 8 or 28, those were the last to go for me… don’t like 8s evidently!)” So, rush right out there and place your bet. It’s 2.10.18.28.30., with powerball 18. Now, don’t ask me what date of the year she’d **never** pick to bet those numbers. She’s a really nice, patient daughter, but I don’t want to push my luck. After all, she’ll be the one deciding whether to put me in the nice nursing home or just lock me in the closet. -Sue ---
**Author:** ![zut](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zut/32/2875_2.png) [@zut](https://boards.straightdope.com/u/zut)\ **Post date:** [October 1, 2001, 4:34pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/33 "2001-10-01T16:34:44Z")
Strictly in the interest of science, I walked down to the Speedway during lunch and invested some time and money in a scheme to verify your daughter’s paranormal anti-powers. As a matter of fact, I contracted the State of Michigan to perform a randomized test, investigating the occurance of the numbers: 1, 10, 18, 28, 30, and **18** as well as, for completeness, 18, 28, 37, 39, 43, and **8** The State will be finishing their testing on Tuesday evening, I believe. I will post here with the results, or (since the State is kind enough to publicly post them) you can check yourself [on their Website](http://www.state.mi.us/milottery/msl_scripts/dr_recent.idc) under the charming description, “The BIG Game.” ---
**Author:** ![jmonster](https://avatars.discourse-cdn.com/v4/letter/j/dbc845/32.png) [@jmonster](https://boards.straightdope.com/u/jmonster)\ **Post date:** [October 1, 2001, 5:03pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/34 "2001-10-01T17:03:24Z")
> [@](#): > > \*Originally posted by Another Primate \* > Now, don’t ask me what date of the year she’d **never** pick to bet those numbers. She’s a really nice, patient daughter, but I don’t want to push my luck. After all, she’ll be the one deciding whether to put me in the nice nursing home or just lock me in the closet. Well, if you were to test your daughter’s powers yourself, you could make sure you’ll be able to afford the nice nursing home. ---
**Author:** ![cornflakes](https://avatars.discourse-cdn.com/v4/letter/c/9d8465/32.png) [@cornflakes](https://boards.straightdope.com/u/cornflakes)\ **Post date:** [October 1, 2001, 5:18pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/35 "2001-10-01T17:18:31Z")
> [@](#): > > \*Originally posted by g8rguy \* > **If you flip a penny 10000 times, it will only be heads about 4950 times, on average; the heads picture weighs more and will hence be more likely to be on bottom. So always guess heads!** Because, as you describe, it will be tails more often and you will win? _Nice try, buddy!_ Actually, my old intro Statistics book gives three instances where people have flipped a coin thousands of times. A Frenchman in the 18th Century got 2048 heads out of 4040 tries for a probability of 50.69% for heads. Karl Pearson, an English statistician, got 12012 heads for 24000 tries (50.05% chance of heads), and English mathematician John Kerrich flipped a coin 10,000 times while imprisoned in WWII Germany and got 5067 heads, or a 50.67% chance. There does seem to be a tiny bias towards heads. Then again, I wouldn’t bet the farm based on three tests either… Now what’s the chance of my remembering something from an old textbook _and_ my being able to find the book when I need it? ---
**Author:** ![Futile\_Gesture](https://avatars.discourse-cdn.com/v4/letter/f/f05b48/32.png) [@Futile\_Gesture](https://boards.straightdope.com/u/Futile_Gesture)\ **Post date:** [October 1, 2001, 5:20pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/36 "2001-10-01T17:20:07Z")
> [@](#): > > \*Originally posted by Another Primate \* > Maybe I was reading her mind to see what she’d ‘call’ next, then flipping the coin _just so_, controlling which side landed up. Yup, that’s it. So, I’m going to apply for the prize! Naah. Just a bizarre coincidence There are ways to fix the way a coin toss ends up. Technical term for this is ‘cheating’ It’s a combination of being very practiced in how you flip the coin and in looking at it as it lands. ---
**Author:** ![DPWhite](https://avatars.discourse-cdn.com/v4/letter/d/d26b3c/32.png) [@DPWhite](https://boards.straightdope.com/u/DPWhite)\ **Post date:** [October 1, 2001, 5:50pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/37 "2001-10-01T17:50:08Z")
Any decent magic shop will be able to sell you a set of coins (one kept in right hand, the other in left) that will help you cheat on coin tosses ---
**Author:** ![g8rguy](https://avatars.discourse-cdn.com/v4/letter/g/7cd45c/32.png) [@g8rguy](https://boards.straightdope.com/u/g8rguy)\ **Post date:** [October 1, 2001, 6:18pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/38 "2001-10-01T18:18:34Z")
> [@](#): > > \*Originally posted by cornflakes \* > Because, as you describe, it will be tails more often and you will win? _Nice try, buddy!_ I was **SO** hoping no one would notice that. Ah, the best laid plans, eh? Now I’m curious, though… Maybe I should devote a few hours to flipping pennies and seeing what happens? ---
**Author:** ![RickJay](https://avatars.discourse-cdn.com/v4/letter/r/bb73d2/32.png) [@RickJay](https://boards.straightdope.com/u/RickJay)\ **Post date:** [October 1, 2001, 6:52pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/39 "2001-10-01T18:52:30Z")
> [@](#): > > \*Originally posted by zut \* > \*\* > > > [@](#): > > > > \*Originally posted by Another Primate \* > > [My daughter] managed to call [a coin flip] wrong 21 times in a row. By the 12th try, she was very upset, but I begged her to persist so we could see how big of a wrong streak she could build up. By 20, she was crying quietly. I wanted her to go on-- she complied once more, started bawling, and I didn’t have the heart to ask her again. > > Wow. Do you realize that the chances of doing that are 1 in two million? I suggest you contact [James Randi](http://www.randi.org/) and see if your daughter can win his million dollar prize. \*\* Actually, no. The odds of incorrectly guessing 21 straight coin flips are one in 67,108,864. An amazing occurrence. You would have MUCH better odds of winning the Lotto 6/49. ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [October 1, 2001, 6:57pm UTC](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046/40 "2001-10-01T18:57:48Z")
> [@](#): > > Actually, my old intro Statistics book gives three instances where people have flipped a coin thousands of times. A Frenchman in the 18th Century got 2048 heads out of 4040 tries for a probability of 50.69% for heads. Karl Pearson, an English statistician, got 12012 heads for 24000 tries (50.05% chance of heads), and English mathematician John Kerrich flipped a coin 10,000 times while imprisoned in WWII Germany and got 5067 heads, or a 50.67% chance. This was in a statistics book? Why would a statistics book make any conclusions based on a measley three trials using as many as three different coins? Seem like bad statistical analysis to me. However, I too wonder how we can ever know if a coin is a fair coin. The more times you flip a fair coin, the lower the probability that you will have a 1:1 ratio of heads to tails. We worked that out right here at the SDMB in this thread: [More Fun With Pure Math](http://boards.straightdope.com/sdmb/showthread.php?threadid=33214) (Most of you probably skipped that one based on the thread title.) Using numbers from the quote above. Flipping a fair coin 4040 times there is only a 1.3% chance of getting 2020 heads and 2020 tails. Flipping a fair coin 24000 times there is only a 0.5% chance of getting 12000 heads and 12000 tails. Flipping a fair coin 1000 times there is only a 0.8% chance of getting 5000 heads and 5000 tails. Even all three trials were done with the same fair coin, there is a 12.2% chance that all three trials would come up with more heads than tails. There is a 24.4% chance that all three trials would show the same bias, either heads or tails. [Previous page](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046.md?page=1) [Next page](https://boards.straightdope.com/t/law-of-averages-and-coin-tosses/84046.md?page=3)