A math connundrum

[JasonEvans]

I was chatting about math and probabilities and the such with my son last night. He just wrapped up his freshman year at Haverford and took soph level Linear Algebra. He got an 800 on his Math SAT and took BC Calc in high school so he is a pretty good math student.

Anyway, we were talking about probabilities and the conversation turned to a place it often down when discussing dice rolls or coin flips -- what happens when the coin is skewing outside of the 50-50 range. In other words, if you have thrown 15 heads out of 20 tosses, is there any enhanced chance that they next toss will be tails to compensate for the run of heads. For those of you who really understand this stuff, we are eliminating the chance that the coin is somehow "spoiled" and inclined to toss heads or tails more than 50% of the time.

I have always asserted that once the event has occurred (meaning the 15 out of 20 has already happened), all future tosses still remain a 50-50 chance. Past events have no impact on future results. My son agreed with me, but then he said that wasn't entirely true. I was sure he was wrong, but he seemed pretty confident of himself. He was convincing enough so that my other son, a junior in high school who just took BC Calc and who got a 790 on the math SAT, agreed with him. I want to see what math experts say about this.

His assertion, in essence, was that over a very, very large number of tosses -- approaching infinity -- a true 50-50 coin will get to 50-50. If the sample size is large enough -- and we are talking about huge numbers here -- then the "law of averages" takes over and forces the coin to reflect its true percentage. The only way it can do that is to correct for the previous imbalanced tosses. So, while each subsequent toss is still a 50-50 chance, over a very large number of tosses, the coin will actually tilt an almost imperceptible amount.

He added the following -- if you were going to toss the coin 5000 times and somehow the first thousand came back with 900 heads and 100 tails, wouldn't you bet that the next 4000 tosses will come out with more than 2000 tails?

Again, I think he is wrong... and maybe I did a poor job explaining his point... but I am dying to know what some of you smart folks have to say about this.

-Jason "by the way, I hope no one thinks I am bragging on my kids talking about the grades, I'm just indicating they are thoughtful about and have a good understanding of math" Evans

[gus]

He added the following -- if you were going to toss the coin 5000 times and somehow the first thousand came back with 900 heads and 100 tails, wouldn't you bet that the next 4000 tosses will come out with more than 2000 tails?

No. This is textbook gamblers' fallacy.

Your son may be misapplying the law of large numbers, which states that as the number of trials increase, the average result will approach the expected value. They key though, is that this does not apply to any small set, and the average approaches the expected simply because the variations become insignificant. In your example, you use a very unlikely event: 900 heads in 1,000 flips of a fair coin. If the next 4000 flips met expectation, you'd have 2,900 of 5,000 flips (58%) coming up heads - significantly closer to expectation than your initial 90%. If you had another 10.000 flips meet expectation, you'd have ~7,900 of 15k (~52.7%).

[CDu]

You are correct. Your son is wrong. The chances of the next toss being heads is still exactly 50%. Over the next 5000 tosses, the epected outcome is 2500 heads and 2500 tails. The coin is without memory.

The idea of large numbers is that as you approach infinity, the observed mean approaches the true expected outcome. That does not mean that, because you have seen tails the 15 of the last 20 tosses that the next 20 should be heads more often.

The idea here is that the error is RANDOM. You don't know when you are going to observe results deviate from the expected outcome. In those next 20 tosses you are just as likely to see more tails again as you are to see more heads.

[Mtn.Devil.91.92.01.10.15]

I think your son is exhibiting observer bias. The individual instances are still 50/50 odds - the past does not have influence on the future.

But perhaps this my old school perspective. I can certainly see his side. And it is quite elevated consideration for a young man. Or young woman.

[sagegrouse]

Again, I think he is wrong... and maybe I did a poor job explaining his point... but I am dying to know what some of you smart folks have to say about this.

-Jason "by the way, I hope no one thinks I am bragging on my kids talking about the grades, I'm just indicating they are thoughtful about and have a good understanding of math" Evans

You are bragging, justifiably so, and both of your sons are wrong on this matter. The ratio will approach 0.50 as the number of trials increase. In fact the std. deviation of the mean value of the ratio will approach zero. But that doesn't mean that there won't be arithmetical differences between the number of heads and tails, and there could be large arithmetical differences that make no measurable differences in the ratio of heads to coin tosses.

"Statistical independence," the concept underlying the measurements, says that each toss has a probability of 0.50, regardless of the tosses that precede it.

"Infinity," a concept not usually used by mathematicians, is really, really big!

[OldPhiKap]

I agree with all us old guys.

The fact that something has a 50-50 chance does not necessarily mean that in any given sample size -- even an extremely large one -- that it actually comes out as many heads as tails.

But I'm a poly sci/history major so take that into account.

[jimsumner]

You are correct.

They are wrong.

The chance of this happening the next time is well under 50 percent.

[duketaylor]

I concur with above rationale, and I enjoy gambling to a small degree

[Deslok]

I concur with above rationale, and I enjoy gambling to a small degree

To put things another way, the ratio of heads to tails will tends towards 1:1, but mathematically if you flip a coin 10000 times and keep track of who is leading, either heads or tails will lead for like 9000 of the flips because one or the other will get out to a fairly small lead in the first 100, and since every flip thereafter is 50:50, that lead of say 55:45 will stay approximately 10 the rest of the way. But winning 55% of the flips is a much greater percentage than winning 5005:4995 but the margin is exactly the same.

As a math teacher, there is a general frustration with how probability and statistics are taught/not taught at a high school level.

[Wander]

He added the following -- if you were going to toss the coin 5000 times and somehow the first thousand came back with 900 heads and 100 tails, wouldn't you bet that the next 4000 tosses will come out with more than 2000 tails?

The probability of flipping a fair coin 1000 times and getting 900 or more heads is far below 1 in a trillion. In fact, I think it is far below 1 in a trillion trillion. It is far below the probability of four 16 seeds in the Final Four. What I'm saying is that I would expect the next 4000 tosses to have far more heads than tails, even though I know you explicitly said it wasn't a weighted coin. I don't believe you

[rsvman]

What they said.

An interesting tangential story is that I once had a mathematician tell me that he used to ask half the class to actually flip a coin 200 times and write down what came up, and the other half of the class to just take a piece of paper and write down heads or tails as though they were flipping a coin, even though they weren't. And he would ask the people who weren't flipping a coin to try to make it look as though they had; in essence, to try to fool him into thinking they had actually flipped a coin.
He said it was really easy to tell who hasn't really flipped a coin, because they would almost never put down a run of heads or tails longer than about 4 or 5, whereas in reality, there will almost always be runs of one or the other that are considerably longer if you actually flip the coin. Students' attempts at randomness are way more ordered than is true randomness.

[ricks68]

Jason wins, son loses. (I have a B.S. in Mathematics from DUKE, and we all know DUKE wins! )

ricks

[BigWayne]

You are correct.

They are wrong.

The chance of this happening the next time is well under 50 percent.

Yes, he has just begun the ascent on the right half of this graph and has a long way to go to get to 50%.:

[madscavenger]

Jason wins, son loses. (I have a B.S. in Mathematics from DUKE, and we all know DUKE wins! )

ricks

rick, i also have a B. S. in Math from Duke (1972). The Department at that time wasn't much to write home about, but at least you didn't have to spend Spring afternoons in a lab. Downside: Both Organic Chemistry and Physical Chemistry, for me, ended up counting as an elective. And yes yes yes, them whippersnappers might be well served to steer clear of casinos.

[BD80]

I was chatting about math and probabilities and the such with my son last night. He just wrapped up his freshman year at Haverford and took soph level Linear Algebra. He got an 800 on his Math SAT and took BC Calc in high school so he is a pretty good math student.

... my other son, a junior in high school who just took BC Calc and who got a 790 on the math SAT, agreed with him. ...

Proof that such scores were more difficult to achieve back when we took the standardized tests.

[TruBlu]

Jason wins, son loses. (I have a B.S. in Mathematics from DUKE, and we all know DUKE wins! )

ricks

Duke gets all the calls, even when flipping coins . . . except in football.

By the way, I lose approximately 90% of coin tosses deciding whether the wife is cooking or we are eating out. Might have something to do with her being in charge of the coin toss while I'm in another room.

[ricks68]

rick, i also have a B. S. in Math from Duke (1972). The Department at that time wasn't much to write home about, but at least you didn't have to spend Spring afternoons in a lab. Downside: Both Organic Chemistry and Physical Chemistry, for me, ended up counting as an elective. And yes yes yes, them whippersnappers might be well served to steer clear of casinos.

Unfortunately, I did the same, as I was also a pre-med student.😨

ricks

[JasonEvans]

Update!!

Older son #1 now says I misunderstood him and he was only saying that over an extremely large set you would have a run of lots and lots of tails to match the run of lots and lots of heads that happened earlier. This is not to "make up for" the previous heads but because abnormal runs are actually to be expected over an extremely large set. He now agrees that once the first set of heads/tails results has happened, there is no greater chance of heads or tails going forward.

It is worth noting that son #1 has never been wrong about anything in his life (or so he believes). My wife and I both think he is altering his argument to fit the overwhelming evidence that his initial theory was false

Son #2 points out that his only point -- which he made during the initial discussion -- was that over a nearly infinite number of chances the early abnormality of more heads than tails becomes insignificant and therefor has an almost meaningless impact on the overall coin results. In other words, if we make 100 throws and are at 80% heads, by the time we get to a quadrillion throws, we will be so close to 50% as to make it appear we made up for the initial 80% imbalance but - in reality -- all that happened was the initial 80% was rendered meaningless by the tremendous number of additional throws.

I never really articulated what son #2 was saying prior to now. I incorrectly stated that he was agreeing with son #1.

-Jason "bottom line, high SAT scores are easier to obtain now than they were in our day" Evans

[OldPhiKap]

Update!!

Older son #1 now says I misunderstood him and he was only saying that over an extremely large set you would have a run of lots and lots of tails to match the run of lots and lots of heads that happened earlier. This is not to "make up for" the previous heads but because abnormal runs are actually to be expected over an extremely large set. He now agrees that once the first set of heads/tails results has happened, there is no greater chance of heads or tails going forward.

It is worth noting that son #1 has never been wrong about anything in his life (or so he believes). My wife and I both think he is altering his argument to fit the overwhelming evidence that his initial theory was false

Son #2 points out that his only point -- which he made during the initial discussion -- was that over a nearly infinite number of chances the early abnormality of more heads than tails becomes insignificant and therefor has an almost meaningless impact on the overall coin results. In other words, if we make 100 throws and are at 80% heads, by the time we get to a quadrillion throws, we will be so close to 50% as to make it appear we made up for the initial 80% imbalance but - in reality -- all that happened was the initial 80% was rendered meaningless by the tremendous number of additional throws.

I never really articulated what son #2 was saying prior to now. I incorrectly stated that he was agreeing with son #1.

-Jason "bottom line, high SAT scores are easier to obtain now than they were in our day" Evans

UGa, Emory, Georgia State, and Mercer all have excellent law schools. Just throwing that out there.

I've never been wrong. But I've been overruled a lot.

[CDu]

Update!!

Older son #1 now says I misunderstood him and he was only saying that over an extremely large set you would have a run of lots and lots of tails to match the run of lots and lots of heads that happened earlier. This is not to "make up for" the previous heads but because abnormal runs are actually to be expected over an extremely large set. He now agrees that once the first set of heads/tails results has happened, there is no greater chance of heads or tails going forward.

It is worth noting that son #1 has never been wrong about anything in his life (or so he believes). My wife and I both think he is altering his argument to fit the overwhelming evidence that his initial theory was false

Son #2 points out that his only point -- which he made during the initial discussion -- was that over a nearly infinite number of chances the early abnormality of more heads than tails becomes insignificant and therefor has an almost meaningless impact on the overall coin results. In other words, if we make 100 throws and are at 80% heads, by the time we get to a quadrillion throws, we will be so close to 50% as to make it appear we made up for the initial 80% imbalance but - in reality -- all that happened was the initial 80% was rendered meaningless by the tremendous number of additional throws.

I never really articulated what son #2 was saying prior to now. I incorrectly stated that he was agreeing with son #1.

-Jason "bottom line, high SAT scores are easier to obtain now than they were in our day" Evans

Son #1 is still wrong, there is no expectation that things will even out later (no matter how many tosses "later" encompasses). Son #2 is correct that the large number principal simply states that over a large enough sample the anomaly does not get corrected, but rather becomes such a small part of the data that it becomes obsolete.

In the 900 of 1000 example, the expected outcome of the next 10,000,000 tosses is 5,000,000 heads. So if that occurs, we see a total of 5,000,900 heads in 10,001,000 tosses, or 50.004%. The mean of the observed data have now approached the true mean without in any way "evening things out" with more tails.