Ok so not really a programming question but not sure where it would go on the stack network...
Dice has 6 sides so rolling a 5 will be a 1/6 (16.6%) chance.
If the dice roll is a 2, does that mean the changes of rolling a 5 now have gone up, and that the changes of rolling a 2 have gone down? Since the odds of rolling two 2's in a row are lower than 16.6%?
Hope that makes sense :)
Each roll is an independent event from another, your chances of rolling any given number on a particular roll, with a fair 6-sided die, is 1/6.
If that explanation isn't good enough... Independence in probability at wiki
Your other question is about the probability of two independent probabilities happening. For example, rolling two 5s in a row. For that you can multiply the independent probabilities together (1/6 * 1/6) to get the probability of doing that (1/36). Three times? 1/216.
A simpler example to think about is coin flipping with a fair coin (50% chance for heads or tails). Does flipping a head one time necessarily mean you are more likely to flip a tail the next? No - you always have a 50% chance of getting either flip next time. Getting two heads in a row? 1/2 * 1/2 = 1/4. Getting a head then a tail? Still 1/2 * 1/2 = 1/4.
No.
Each throw of the dice is an independent event. The probability remains the same.
If you are interested in learning about probability and statistics, take a look at Head First Statistics
Every possible outcome is indepent from the previous one. This means that with one roll you will be have 16.6% (1/6) for every combination 1,2,3,4,5,6.
With two rolls you will have 0.027% (1/6 * 1/6 = 1/36) for every combination 11, 12, 13, ..., 31, 32, 33, ... 64, 65, 66
No. Not at all. The odds of rolling 2 2s is low but so is the odds of rolling 2 the first time and 5 the second time.
In most cases successive dice rolls are considered independent events, hence outcome of one roll does not affect the outcome of another.
The probability of getting three rolls of 5 in a row is small. But this does not mean that your chances during that third roll are changed. They are still 1 out of 6.
http://en.wikipedia.org/wiki/Independence_(probability_theory)
An interesting fact, though is that you can take independent events, and make them somewhat dependent by grouping them:
The trick here is that you are calculating that three die rolls will be certain values. They're still independent, but you have grouped them, so you must combine their odds. It may be easier to think about this if you roll three dice at the same time, rather than one after the other.
To believe anything other than the fact that each roll is statistically independent of the previous rolls (e.g. the thought that a roll is more likely to come up because it hasn't come up previously) is to fall into the Gambler's Fallacy.
You are always throwing a second time ... someone rolled the dice before you. In yor experiment every roll is independent from the previous one. That's not always the case in every experiment: if you are getting balls of two colors out of a bag without reposition each succesive round is conditioned by the previous results!
As mentioned, every throw of a die is independent, however, let's make this a bit more interesting!
Let's say that on your first roll you do not roll a 5. There is a 5/6 or 83.3% chance of this happening. You can see this by subtracting the chance of rolling a 5 from 1 like so:
1 - 1/6 = 5/6
There would further be a 69.4% chance of not rolling a five on the second roll:
(5/6)^2 = .694
Or another way of saying it is that there is a 30.6% (1 - .694) chance of rolling a 5 in the first two rolls of a die.
If you continue this pattern up to 38 rolls of a die then you reach:
1 - (5/6)^38 = .999
Meaning that if you roll a single die 38 times there is a 99.9% chance that you will roll a 5.
So if you roll a single die 38 times without rolling a 5, then surely there's no way the next roll won't be a 5!
There. Glad that's all cleared up then.
Do you seriously believe that the die "remembers" it rolled a 2 last time, and has now "decided" to increase the odds of rolling a 5?
As soon as you think about what would be involved in a die "remembering" its previous rolls and influencing its own future, you'll realize that your hypothesis requires a plastic die to be alive, with memory, thoughts, and free-will.
Clearly that is absurd. The odds do not change from one roll to the next. (if they did, I could go make a killing in Las Vegas).