Calculating Probability of Rolling Same Number Twice with One Die in 3 Rolls

[hadron23]

Hello,

This is probably an extremely simple question and my lack of sleep is preventing me from understanding it completely, but I figured I would ask.

Suppose you have one fair die. You roll it 3 times. What is the probability that the first and second roll are the same number? My guess is that it is 1/6; however, I was also thinking about how to solve the problem with the frequentist approach.

We know that there are 6^3 = 216 possible combinations of a 6 sided die in 3 rolls. Given that there are only six possible combinations where the numbers of the first and second roll can be the same (1-1,2-2,3-3,4-4,5-5,6-6), I would think the probability (using this approach) would be 6/216. However, I think this is clearly wrong, since by increasing the number of rolls to say 5 (so the total number of combinations is 6^5 = 7776), the same method would say the probability of the first two numbers being the same is now 6/7776 (clearly does not make sense!).

What am I overlooking? Are there underlying assumptions of the frequentist approach that I am not considering? Clearly, the probability of the same number of the first and second roll should be independent of how many rolls we perform (beyond 2 rolls).

Thanks

Edit: Err...title should read "One Die" *embarrassed*

 

[rollcast]

Try constructing a probability tree diagram so you can see it visually.

Theres no need for 6 branches just call the one x and the other branch not x.

 

[rollcast]

Also surely the 3rd roll of the dice doesn't influence the probability of the first 2 rolls so there is no need to include them in your solution.

Therefore I would be rereading the problem as what's the probability of rolling a dice 2 times and getting the same number.

Unless I missed something.

 

[pbuk]

Yes there are 216 combinations. How many of them are 1-1-X ? With 5 rolls, how many are 1-1-X-Y-Z? With N rolls, what fraction of N are 1-1-X₃-...-X_N?

 

[HallsofIvy]

In order that the first two rolls be the same, the first roll can be anything, then the second roll must match it. Since the first roll be any of the 6 values, the probability that the second roll matches it is 1/6, as you say. And, as roll cast says, the third roll is irrelevant.

 

[tiny-tim]

hello hadron23!

… We know that there are 6^3 = 216 possible combinations of a 6 sided die in 3 rolls. Given that there are only six possible combinations where the numbers of the first and second roll can be the same (1-1,2-2,3-3,4-4,5-5,6-6),…

no, there's 36 combinations (1-1-1 to 1-1-6, 2-2-1 etc, 3-3-1 etc, …)

since I'm basically repeating what everyone else has said, I've really only joined in so that i can remark …

Edit: Err...title should read "One Die" *embarrassed*

die of embarrasment!

 

[chiro]

Hello,

This is probably an extremely simple question and my lack of sleep is preventing me from understanding it completely, but I figured I would ask.

Suppose you have one fair die. You roll it 3 times. What is the probability that the first and second roll are the same number? My guess is that it is 1/6; however, I was also thinking about how to solve the problem with the frequentist approach.

We know that there are 6^3 = 216 possible combinations of a 6 sided die in 3 rolls. Given that there are only six possible combinations where the numbers of the first and second roll can be the same (1-1,2-2,3-3,4-4,5-5,6-6), I would think the probability (using this approach) would be 6/216. However, I think this is clearly wrong, since by increasing the number of rolls to say 5 (so the total number of combinations is 6^5 = 7776), the same method would say the probability of the first two numbers being the same is now 6/7776 (clearly does not make sense!).

What am I overlooking? Are there underlying assumptions of the frequentist approach that I am not considering? Clearly, the probability of the same number of the first and second roll should be independent of how many rolls we perform (beyond 2 rolls).

Thanks

Edit: Err...title should read "One Die" *embarrassed*

Hey hadron23.

This kind of things happens more than you think, even among mathematicians.

With probability we can use assumptions to derive probability measures. In the dice assumptions we are assuming that every roll is independent of each other. This assumption simplifies the whole scenario.

Now there is nothing to stop us to make a model based on say classical mechanics and while that may yield answers to an extremely high accuracy, it is for most purposes, way too complicated: for this reason we use the probabilistic model that incorporates indepedence.

Also you should be aware of the reverse process to the above. In other words given a sample we estimate the probability of getting a specific value of say a dice giving us a 1, or a coing giving us heads.

To give you an example imagine we get data for a set of coin flips giving us a hundred heads in a row followed by two tails. From this data, we might be asked to estimate the probability of getting a head and getting a tail.

Using this data, we can build confidence intervals for the probabilities and we find that the probability for getting a head is extremely unbiased (way bigger than 0.5). However it may turn out that after rolling a million or a billion rolls, that the results end up giving us an expected result of 0.5, but for this sample we don't know that and using standard statistical techniques we get something that is very different.

As you can see, this presents a very big problem. Probability has different interpretations and if you end up studying it in depth you'll find that you come across more of these kinds of problems.

The best we can do is to use mathematical results and assumptions and try to put a process into context of those assumptions. But even then we come across situations where probability is based on data and 'gut feelings' and other subjective values which are not really explainable in sufficient detail or understanding and this affects many areas of science and also areas of philosophy.