Explain this statistics/probability thing for me

[Bone_Vulture]

The thing is, on most lotteries, they don't have to come up "in order". If that were required, the odds of winning would be... (what's larger than "astronomical"?) The numbers drawn could be 5-3-2-4-1 and you would win.

But I see your point -- they always come out in numerical order on your ticket.

Oh yeah, I just recalled that according to some newspaper article I read a long while ago, the most popular combination that people play on the national lottery is 1-2-3-4-5-6-7. Now, even with my meager knowledge on statistics, this sounds unbelievably stupid, as such a linear combination is surely less probable than a more random pick, not to mention that even if you managed to win, you'd have to share the jackpot with probably hundreds of other true believers.

 

[tsg]

Ok, I did some calculations: the Finnish lottery has 39 numbers, jackpot requires seven correct picks. I will assume that the absolute probability of guessing the combination correctly at one try is 39 times 38 times [...] 33, which adds up to 77,519,922,480.

If the Finnish lottery requires you to pick the winning numbers in order, then that is correct. If order doesn't matter, it's one in 15,380,937.

 

[Bone_Vulture]

In simplest terms, the coin doesn't know what it did previously. The law of averages just means that, in the long run, the probabilities will average out. That is, in large enough samples the occurrence of random events will be approximately equal to the probability and that the larger the sample, the closer to average the outcome is likely to be. Tossing a coin five times and not getting any tails is unlikely, but no less likely then any other particular string of tosses (for example HHTTH). That it has happened doesn't influence what will happen next. HHHHHH is just as likely as HHHHHT.

I read more on Wikipedia, the statistical theory is becoming more clear now. Can you still comment on my lottery assumption?

[EDIT]

Ah, thanks. Oh yeah, correct. I forgot to remove that factor from the formula, I already started thinking that 77 billion is so insanely small that no-one would ever win in this country, with a population of five million.

 

[gnome]

An interesting argument. I am aware that the absolute probability remains at the 50/50 in this coin toss example - although I still don't understand why a person cannot trust on the cumulative probability that'll probably "even things out".

You can... but the probability over time increases at the same rate whether you change numbers or not.

 

[tsg]

Oh yeah, I just recalled that according to some newspaper article I read a long while ago, the most popular combination that people play on the national lottery is 1-2-3-4-5-6-7. Now, even with my meager knowledge on statistics, this sounds unbelievably stupid, as such a linear combination is surely less probable than a more random pick, not to mention that even if you managed to win, you'd have to share the jackpot with probably hundreds of other true believers.

1-2-3-4-5-6-7 is no less likely than any other 7 number combination. Although the reasoning for not playing due to the number of others that play it is quite correct.

 

[Bone_Vulture]

If the Finnish lottery requires you to pick the winning numbers in order, then that is correct. If order doesn't matter, it's one in 15,380,937.

To tell a bit of my background, I'm supposed to graduate from a business polytechnic within a year, so obviously fumbling like this with statistical math is a bit of an embarrassment. So, I decided to redeem myself by figuring the correct formula that'll arrive to that number. It's 39/7 times 38/6 times [...] 33/1. To learn is to find solutions to the mistakes one makes.

 

[Schneibster]

You're making this up.
~~ Paul

Try it for yourself. Point out that the odds of throwing three heads in a row are 1/2*1/2*1/2=1/8, implying 7:1 against, and you've just thrown two heads, thus the odds of throwing a third head are only one in eight. State that you only want 2:1 odds; if it's tails, you pay them a dollar, if it's heads, they pay you two.

Drink beer.

ETA: There's a sucker born every minute, and as noted above, many people will bet heavily during a losing streak, because "the odds can't go against me forever."

 

[Bone_Vulture]

1-2-3-4-5-6-7 is no less likely than any other 7 number combination. Although the reasoning for not playing due to the number of others that play it is quite correct.

Correct, I guess it's the superstition factor again. I'm content with seeing that the seven numbers are [any random set] weekly, but seeing the exact 1 through 7 combination ever coming up would really make me drop my proverbial monocle.

It's just as unlikely as any other combination, only more - er, dramatic. I think the final exercise would be to figure the odds for seeing this combination during my lifespan, which'll be some.. well, fifty is well enough.

 

[Bone_Vulture]

Try it for yourself. Point out that the odds of throwing three heads in a row are 1/2*1/2*1/2=1/8, implying 7:1 against, and you've just thrown two heads, thus the odds of throwing a third head are only one in eight. State that you only want 2:1 odds; if it's tails, you pay them a dollar, if it's heads, they pay you two.

Drink beer.

The odd thing is that I would refuse the bet, unless the two "preliminary heads" were thrown only after the bet is agreed upon. Perhaps I'm sucker then.

Another memory: when I was younger, I used to play pen & paper RPG's, that involved throwing a 20-sided die a lot. A friend's friend had an amusing habit of throwing his die idly while waiting for his turn, trying to get any other number than 20, stating that the odds for the ideal number would be higher when he'd make the actual throw. Go figure.

 

[Soapy Sam]

The OP describes a classic case of the "Bus Stop Effect".

Put anyone at a bus stop and keep him there twelve minutes, by force if necessary. He will now be convinced that a bus will be along in a minute.
He cannot leave the bus stop, as he knows that as soon as he does so, the bus will arrive.
I proposed this method to HM Prison service several years ago as a solution to overcrowding. It worked so well that three out of any ten people in a given bus queue are now convicted felons. So watch your money.

The UK Lottery knows very well that many people choose the same numbers (often birthdays, so sub-31) each week. That was why they started doing two draws a week- because everyone KNOWS that their numbers will come up on a Wednesday if they only buy tickets for Saturday.

It's like printing "Rinse and repeat" on shampoo bottles. Sales doubled by three words.

 

[tsg]

Correct, I guess it's the superstition factor again. I'm content with seeing that the seven numbers are [any random set] weekly, but seeing the exact 1 through 7 combination ever coming up would really make me drop my proverbial monocle.

It's just as unlikely as any other combination, only more - er, dramatic. I think the final exercise would be to figure the odds for seeing this combination during my lifespan, which'll be some.. well, fifty is well enough.

Well, the odds of it not coming up in one game are 15,380,936 / 15,380,937 or 0.999999934984455. So, assuming one game per week, the odds of the number not coming up in 50 years is 0.999999934984455^(50*52) = 0.999830974. That makes the odds of it coming up in 50 years 1 - 0.999830974 = 0.000169026 or roughly 1 in 5916. This is assuming Excel hasn't made any rounding errors.

The next trap is that this is only true for the specific sequence 1-2-3-4-5-6-7 and not just any sequence that makes a recognizable pattern. If you're going to count 2-3-4-5-6-7-8, 3-4-5-6-7-8-9, 2-4-6-8-10-12-14, 3-6-9-12-15-18-21 etc., the odds are significantly higher that you will see a winning set of numbers that forms an interesting pattern.

 

[Bone_Vulture]

And finally: Wikipedia explains the psychology behind the discussed behavior in a section titled "Gambler's fallacy". Choice quote:

You are on a game show. You are given a choice between three doors. Behind one door is a new car; behind the other two there are goats. You pick a door. The host reveals a goat behind one door that you didn't pick. Now there are two remaining doors. The host gives you the option of reconsidering your choice. Should you take the host's offer? Answer: yes you should; it doubles your chances of winning the car. This is the famous Monty Hall problem, and it is a great example of how unreliable intuition can be when considering even simple probability problems.

I remember this example from an episode of Numb3rs. Maybe I should start watching that again?

 

[Melendwyr]

An interesting argument. I am aware that the absolute probability remains at the 50/50 in this coin toss example - although I still don't understand why a person cannot trust on the cumulative probability that'll probably "even things out".

The record of what you actually get from flipping coins will approach the theoretical probability (which we'll assume is 50/50) as the number of coin flips approaches infinity. That's called the "Law of Large Numbers".

But here's the catch: the record is only guaranteed to approach the theoretical value, not reach it or balance around it. Let's say you have an unusual streak of heads, followed by a perfectly equal mix of heads and tails. As you flip more often, the blip of heads will have a smaller and smaller effect on the record. Eventually it's negligible.

By and large, a temporary bias towards heads or tails will tend to be compensated by another temporary bias towards the other, but it's not inevitable. Each coin flip is independent of every other coin clip - the coin doesn't know that it needs to "balance out" its past behavior.

Does that make sense?

 

[Schneibster]

The odd thing is that I would refuse the bet, unless the two "preliminary heads" were thrown only after the bet is agreed upon. Perhaps I'm sucker then.

Well, as long as you require that the three heads be thrown in a row as the first three throws after the bet was made, you'd win over the long run- but nobody would bet with you if they had any brains.

Another memory: when I was younger, I used to play pen & paper RPG's, that involved throwing a 20-sided die a lot. A friend's friend had an amusing habit of throwing his die idly while waiting for his turn, trying to get any other number than 20, stating that the odds for the ideal number would be higher when he'd make the actual throw. Go figure.

Heh, yep. Same principle.

 

[Bone_Vulture]

Does that make sense?

As I read it earlier, your explanation was echoed on Wikipedia almost word to word. But thank you for including the explanation to the thread directly as well.

Well, as long as you require that the three heads be thrown in a row as the first three throws after the bet was made, you'd win over the long run- but nobody would bet with you if they had any brains.

And that's where the liquor comes in! These days, I'm nearly stone sober, so with a sales pitch of "another beer for you or twice the beer's price in euros for me" would most likely work in my benefit in the long run.

 

[GreedyAlgorithm]

By and large, a temporary bias towards heads or tails will tend to be compensated by another temporary bias towards the other, but it's not inevitable.

Everything else you said was correct, but this is the sort of sentence that leads otherwise rational people into the Gambler's Fallacy. A temporary bias towards heads or tails WILL NOT tend to be compensated by another temporary bias towards the other. The only reason temporary biases are temporary is because of the large sample size effect, namely, that a bias in 100 throws does not show up significantly after an unbiased 10,000 throws. If you get 70 H and 30 T, but then get 5026 H and 4974 T, then the first 100 throws were heavily biased toward H, there wasn't a temporary bias toward T, but the 10,100 throws are only imperceptibly biased toward H.

 

[tsg]

And finally: Wikipedia explains the psychology behind the discussed behavior in a section titled "Gambler's fallacy". Choice quote:

You are on a game show. You are given a choice between three doors. Behind one door is a new car; behind the other two there are goats. You pick a door. The host reveals a goat behind one door that you didn't pick. Now there are two remaining doors. The host gives you the option of reconsidering your choice. Should you take the host's offer? Answer: yes you should; it doubles your chances of winning the car. This is the famous Monty Hall problem, and it is a great example of how unreliable intuition can be when considering even simple probability problems.

I remember this example from an episode of Numb3rs. Maybe I should start watching that again?

This one is always good for an argument.

 

[Interesting Ian]

The UK Lottery knows very well that many people choose the same numbers (often birthdays, so sub-31) each week. That was why they started doing two draws a week- because everyone KNOWS that their numbers will come up on a Wednesday if they only buy tickets for Saturday.

The Wednesday draw was another stupid idea. It pales into insignificance compared to the lucky dip though.

 

[Ziggurat]

You can pick 1-2-3-4-5 as your five numbers one week. Someone might tell you, "No way that's ever going to happen, all the numbers are going to come up in order? Never." But it is just as likely to happen as any other possible, more random-looking sequence.

People are very bad at being intentionally random.

I remember hearing some anecdote about some professor doing a "test" to see if his class was psychic. He told then he'd picked out a sequence of something like 8 random 0's and 1's, and had the class try to guess what he had written down. He had them all raise their hands, he'd start calling out the numbers, and they'd lower their hands once they had gotten one wrong. By statistics, you'd expect half the hands to go down each time, but it didn't. After half the numbers had been read out, which was 1011, a much larger number than you might expect still had their hands up. Aha! the class was psychic.

Then the professor read the rest of the numbers, 0000, and the remaining hands went down quickly. Their psychic abilities failed.

What happened? Well, if you ask people to pick a random series of 4 1's and 0's, they don't produce a random list. Almost nobody picks 0000 or 1111. 1011 and 0100 are very common, since people start out alternating, but then decide they need to do some repeats because repeats sometimes happen. The net results of people's "random" picks are statistically very skewed, and all the professor did was take advantage of that fact.

As a side note, it is possible to train people and even animals to act more randomly.

 

[gnome]

And finally: Wikipedia explains the psychology behind the discussed behavior in a section titled "Gambler's fallacy". Choice quote:



I remember this example from an episode of Numb3rs. Maybe I should start watching that again?

AAAAAAAAHHHHHHHHHHHHHHH!!!!!

This thread will soon be about 12 pages long.