probability question

[Special K]

I was once playing a game in which each player rolls a standard six-sided once die during their turn. At one point during the game, someone pointed out that they were "bound to roll a 6 this time, because no one had rolled one so far in the game", and the game had been going on for quite some time. Another player responded by saying "the events are independent, previous rolls don't have any effect on the probability of you rolling a 6 this turn."

While I believe the second player's response is correct, what bothers me is this:

Successive rolls of a die are independent events. The probability of rolling a 6 is 1/6, regardless of what has happened in the previous rolls. However, if you look at all the dice rolls that had occured in the game up to that point, the probability of none of them being a 6 is quite low (assuming the number of rolls is high enough), and therefore if you consider the next roll coming up as part of the whole, it seems even more unlikely that it wouldn't be a 6, because the probability of that many consecutive rolls not being a 6 would be incredibly low. Eventually the probability of a number of consecutive rolls without a 6 would become so low it would seem the next roll would almost 'have' to be a 6.

So how do you reconcile these two perspectives of what seem to be independent events?

 

[BrownTown]

dice roles are independant events, do there is a 1/6 chacne you will get a 6 regardless of what the other rolls are (assuming the dice is fair). So, yeah it might be unlikely that nobody roles a 6 in say 20 turns, but if you keep rolling for say 100 turns then it might turn out that you get 4 sixes in a row, its jsut pure luck. As the number of roles increases the percent of sixes roles should get closer to 1/6. However, if you role 20 times and get no sixes and then trole 80 more times you would not expect to get 1/6, but instead .8/6 since the first 20 events are already determined, and will not affect the outcome of the last 80.

 

[blahblah99]

Originally posted by: Special K
I was once playing a game in which each player rolls a standard six-sided once die during their turn. At one point during the game, someone pointed out that they were "bound to roll a 6 this time, because no one had rolled one so far in the game", and the game had been going on for quite some time. Another player responded by saying "the events are independent, previous rolls don't have any effect on the probability of you rolling a 6 this turn."

While I believe the second player's response is correct, what bothers me is this:

Successive rolls of a die are independent events. The probability of rolling a 6 is 1/6, regardless of what has happened in the previous rolls. However, if you look at all the dice rolls that had occured in the game up to that point, the probability of none of them being a 6 is quite low (assuming the number of rolls is high enough), and therefore if you consider the next roll coming up as part of the whole, it seems even more unlikely that it wouldn't be a 6, because the probability of that many consecutive rolls being a 6 would be incredibly low. Eventually the probability of a number of consecutive rolls without a 6 would become so low it would seem the next roll would almost 'have' to be a 6.

So how do you reconcile these two perspectives of what seem to be independent events?

These two "perspectives" come from the fact that you know the history of the rolls. Suppose your friend tallies up each dice roll from the start while you just join in on the game after 50 rolls. To you, the probability of rolling a 6 is 1/6 because your history starts at the 50th roll, while to your friend it the history started at roll 0. His probability of getting a 6 with history included may not be 1/6th, but as the number of rolls reaches infinity, both of you will reach 1/6.

 

[Armitage]

Every individual roll is independent - the history of rolls to that point can't possibly have any impact. Ask yourself what the physical mechanism would be such that the past rolls could influence the current roll.

Also note that any series of dice rolls has exactly the same chance of occuring - so it is just as likely that you would roll

6 6 6 6 6 6 6 6 6 6

as

1 4 2 1 6 3 2 5 1 2

You would think that the first case was amazing - but wouldn't think twice about the second. It's all in your head

 

[Armitage]

Originally posted by: blahblah99

Originally posted by: Special K
I was once playing a game in which each player rolls a standard six-sided once die during their turn. At one point during the game, someone pointed out that they were "bound to roll a 6 this time, because no one had rolled one so far in the game", and the game had been going on for quite some time. Another player responded by saying "the events are independent, previous rolls don't have any effect on the probability of you rolling a 6 this turn."

While I believe the second player's response is correct, what bothers me is this:

Successive rolls of a die are independent events. The probability of rolling a 6 is 1/6, regardless of what has happened in the previous rolls. However, if you look at all the dice rolls that had occured in the game up to that point, the probability of none of them being a 6 is quite low (assuming the number of rolls is high enough), and therefore if you consider the next roll coming up as part of the whole, it seems even more unlikely that it wouldn't be a 6, because the probability of that many consecutive rolls being a 6 would be incredibly low. Eventually the probability of a number of consecutive rolls without a 6 would become so low it would seem the next roll would almost 'have' to be a 6.

So how do you reconcile these two perspectives of what seem to be independent events?

These two "perspectives" come from the fact that you know the history of the rolls. Suppose your friend tallies up each dice roll from the start while you just join in on the game after 50 rolls. To you, the probability of rolling a 6 is 1/6 because your history starts at the 50th roll, while to your friend it the history started at roll 0. His probability of getting a 6 with history included may not be 1/6th, but as the number of rolls reaches infinity, both of you will reach 1/6.

Eh??? The chance of rolling a 6 is 1/6 regardless of your "perspective" or any history of past rolls.

This is all assuming the die is true of course - any chance you were rolling an unusual number of 1s?

 

[krotchy]

Probability theory generaly looks at everything as if it is an independent event. Statistics theory likes to look at the whole picture more. It tends to include prior stats to make determinations. So while the probability of a 6 is always 1/6. If you havent rolled a 6 in a long time statistically, you are more likely to get a 6.

Think of it like an NBA free throw shooter. If he shoots 70% over his career, statistically in a game where he is 6/12 he will make the next free throw to converge his % closer to 70% from 50%. According to probability however he has a 70% shot of making it. They are the same field of math but different in essence I suppose.

Statistics vs Probability. the same field and different fields at the same time. Hah.

 

[DrPizza]

Originally posted by: Armitage
Every individual roll is independent - the history of rolls to that point can't possibly have any impact. Ask yourself what the physical mechanism would be such that the past rolls could influence the current roll.

Also note that any series of dice rolls has exactly the same chance of occuring - so it is just as likely that you would roll

6 6 6 6 6 6 6 6 6 6

as

1 4 2 1 6 3 2 5 1 2

You would think that the first case was amazing - but wouldn't think twice about the second. It's all in your head

relatedly, the probability of the lotto multi-mega bazillion jackpot numbers being
1 2 3 4 5 6 with super special additional number 7
are just as likely as any other combination. But, people still spend billions of dollars nationally on the lottery.

 

[Armitage]

Originally posted by: krotchy
Probability theory generaly looks at everything as if it is an independent event. Statistics theory likes to look at the whole picture more. It tends to include prior stats to make determinations. So while the probability of a 6 is always 1/6. If you havent rolled a 6 in a long time statistically, you are more likely to get a 6.

Absolutely not, no matter what you call it. Statistics and probability are inextricably intertwined - there is no meaning to saying probability vs. statistically.

Think of it like an NBA free throw shooter. If he shoots 70% over his career, statistically in a game where he is 6/12 he will make the next free throw to converge his % closer to 70% from 50%. According to probability however he has a 70% shot of making it. They are the same field of math but different in essence I suppose.

Statistics vs Probability. the same field and different fields at the same time. Hah.

Uhm, no - an independent event is still an independent event. There is no such thing as one particular outcome of an independent event being "due" - unless you're a sports commentator

Although, free throw shooting is likely not a series of independent events - not becuase the shooter is statistically "due" to make a shot, but because of the various physiological and psychological factors that come into play.

 

[bobsmith1492]

You can't look to the past like that; you have to look to the future.

What I mean is you cannot say based on past results (with a system in which the probability of a particular even is determined, in this case, 1/6 chance every time) what your next result is going to be.

The way you are thinking is like a (oh, I forgot - geometric series?) in which as the number of trials goes to infinity, the probability of a particular outcome occuring goes to 1.

So, what you COULD say is, "If I roll 10 more times, I have a XX chance of getting a 6," where XX would be much greater than 1/6. The same cannot apply to your next individual roll, however. *

* - Actually, it does... the probability equation I'm thinking of is of the form 1-(1-p)^n, where n is the number of trials (dice rolls) you will attempt, and p is the probability of the desired event. If you're looking only at your next roll (n=1), you'd get 1-(1-p), or p, so in this case, p=1/6, so you would still have a 1/6 chance of rolling whatever.

As you can see from this, your "probability" and "statistics" line up exactly. It's when n gets larger that you see differences. If n=5, you would still have a 1/6 chance of rolling some number on each roll, but from 5 rolls, you would have a 1-(5/6)^5 chance, or 59.8% chance of rolling at least one of your particular number.

That does NOT imply that if you haven't rolled your number after 4 tries, you have a 59.8% chance of rolling a 6. If you say that, you've effectively reset your n count to 1 ("The next roll" is the keyword). I believe this is called memory, which this does not exhibit at all.

That is one of the biggest mistakes to be made in probability/statistics. Hopefully that helped some.

EDIT: This was to blahblah, krotchy, and the OP. The other guys seem to be on track.

 

[randay]

Dice rolls are never truly random. the position of the dice in your hand, the speed of your roll, the speed of the spin, the rolling surface, and the shape and composition of the dice will almost always certainly change the ideal 1/6 probability of any single number landing face up. I know this isnt what the arguement is about but it could explain your particular scenario.

 

[joshd]

Well, it is unlikely that they would all be rolling in a particular way.

Yes, the events are all independant. it does not matter what has gone before, as this cannot affect it. As Armitage said, the event of a 6 cannot be "due" if the events are truely independant of each other.

 

[CycloWizard]

Well, everyone else has already addressed the technical aspects of the answer, so I'll go with the logical portion. Your friend's conclusion that you're due for a 6 any time is actually known as the Gambler's Fallacy in logic. There are lots of interesting examples of how this can be applied if you google it.

 

[Special K]

Alright, I read the wiki article on the Gambler's Fallacy, which does a good job of summarizing what others have been posting. However, that leads me to another question - how can we ever speak of the probability of say, 5 coin tosses coming up heads? As soon as we throw the first coin, the event is no longer unknown, and doesn't factor into the "experiment" anymore. If all of the coin tosses are independent, how does it even make sense to speak of the probability of "tossing 5 heads in a row"? As soon as we toss the first one, it's not the same experiment anymore.

 

[joshd]

yea... well once you have one head, then the next toss you are looking for another head, but it cant be realated to the previous one, as the previous one cannot affect it. it is NOT the same as taking coloured balls out of a bag, and NOT putting them back in before choosing your next ball. the last toss doesnt influence the next.

to work out the probability of 5 heads, you work out 0.5^5. with the ball one, every time you draw a ball, the probabilities change, as there is a different number of balls in the bag.

 

[BucsMAN3K]

You can account for the probability that you would get only one 6 in so many rolls, which would be less the more rolls there are. So overall you could say that it is very improbable that you would NOT roll a 6, but you probability of rolling a 6 when just considering that single event is still 1/6th.

So really, it's all relative to what you are considering, because if you think about it, in light of everything that could happen in the universe, what the probability that ANYTHING happens?

 

[edcarman]

Originally posted by: Special K
how can we ever speak of the probability of say, 5 coin tosses coming up heads? As soon as we throw the first coin, the event is no longer unknown, and doesn't factor into the "experiment" anymore. If all of the coin tosses are independent, how does it even make sense to speak of the probability of "tossing 5 heads in a row"?

The second part of this is what is known as conditional probability. Basically it is the probability of some event occuring given that some other event has already occured.

The first part (probability of 5 heads) is rather like telling your friend to go away, toss a coin five times and then come back and tell you the result. You are predicting the chance that he will come back and tell you he got 5 heads - a probability of (0.5)^5 or 1 in 32.

The second part is like telling him to go away and toss the coin five times, but this time he tells you the result of each toss and you update your estimate based on the result:

If he tells you that the first toss was a tail then, obviously, the probability of getting five heads is now zero. The probability of 5H in 5 throws given that the first throw is T is zero.

If he tells you that his first four throws were all heads, then you can predict the probability of him now getting 5 heads altogether is 0.5 (the probability that his last throw is a head). The probability of 5H in 5 throws given that the first four are HHHH is 0.5.

Basically, the probability of any individual coin toss turning up heads is still the same, but the overall probability you are calculating is changing.

 

[Armitage]

Originally posted by: edcarman

Originally posted by: Special K
how can we ever speak of the probability of say, 5 coin tosses coming up heads? As soon as we throw the first coin, the event is no longer unknown, and doesn't factor into the "experiment" anymore. If all of the coin tosses are independent, how does it even make sense to speak of the probability of "tossing 5 heads in a row"?

The second part of this is what is known as conditional probability. Basically it is the probability of some event occuring given that some other event has already occured.

The first part (probability of 5 heads) is rather like telling your friend to go away, toss a coin five times and then come back and tell you the result. You are predicting the chance that he will come back and tell you he got 5 heads - a probability of (0.5)^5 or 1 in 32.

The second part is like telling him to go away and toss the coin five times, but this time he tells you the result of each toss and you update your estimate based on the result:

If he tells you that the first toss was a tail then, obviously, the probability of getting five heads is now zero. The probability of 5H in 5 throws given that the first throw is T is zero.

If he tells you that his first four throws were all heads, then you can predict the probability of him now getting 5 heads altogether is 0.5 (the probability that his last throw is a head). The probability of 5H in 5 throws given that the first four are HHHH is 0.5.

Basically, the probability of any individual coin toss turning up heads is still the same, but the overall probability you are calculating is changing.

Well said :thumbsup:

 

[theMan]

each dice roll is independent, but, the probability of rolling the dice 100 times, and not getting a 6 is different.

 

[BrownTown]

maybe its just me, but why does it seem every simple question in Highly Technical ends up with people making huge explanations, or people trying so hard to complicate an easy thing that they end up confusing everyone?

Anyways I probably shouldn't ask that sicne now ill probably get a 3 page lecture on psychology explaining why people try to overcomplicate easy ideas...