Probability of 6 or 1 with Two Dice: An Infinite Experiment

In summary, the probability of getting either a 1 or 6 on a single roll of a die is 1/3. When rolling two dice and only wanting a 6, the probability of getting at least one 6 is 11/36. However, if the two 6s are counted as 2, then the probability is 2/3. When throwing the dice an infinite number of times, the expected number of 1s and 6s on the first die and the expected number of 6s on the second die will be the same, as shown by the principle of indifference and the law of large numbers. This can also be calculated using the expected value for a random variable. Additionally, it
  • #1
aaaa202
1,169
2
Suppose I throw a dice and want either 6 or 1. The probability for that is obviously 1/3. Now suppose instead, that I throw two dices but only want 6 this time. Then the probability of getting 6 is a bit lower than 2/3, but instead there is of course also a probability of getting for instance 6 on both dices.

Now say I throw the dices an infinite number of times. Will the average number of 1's and 6'1 on the first dice and average number of 6's on the 2nd two dices be the same?

And if so, how can I realize that?
 
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  • #2
If you count the two die 6-6 case as 2 then yes.
 
  • #3
By the way, there is no such word (in English) as "dices". "Dice" is the plural of "die".
 
  • #4
You can try using the expected value for a random variable.

Halls of Ivy: Congratulations on your 2^15 posts!.
 
  • #5
Bacle2 said:
You can try using the expected value for a random variable.

Halls of Ivy: Congratulations on your 2^15 posts!.

You could, but I think that's really overkill. The only difference between the two situations is the labeling of the sides. The two cases must give the same result by a principle of indifference.
 
  • #6
kai_sikorski said:
You could, but I think that's really overkill. The only difference between the two situations is the labeling of the sides. The two cases must give the same result by a principle of indifference.

You're right that it's overkill; I just thought the OP asked to have a formal proof.
 
  • #7
Someone once told me it had to do with the fact, that both of them follow a "product distribution" or something like that? Is that true?
 
  • #8
btw... It seems like you guys think of it as intuitive. I don't - please give me an example that makes it intuitive :)
 
  • #9
HallsofIvy said:
By the way, there is no such word (in English) as "dices".
Be careful; you're dicing with death with that remark. :wink:
 
  • #10
aaaa202 said:
btw... It seems like you guys think of it as intuitive. I don't - please give me an example that makes it intuitive :)

Okay, maybe the thing to do is to go to expectations. Let X1 be 1 if the die is 1 or 6. Due to mutual exclusion of those events and law of total expectation

E[X1 | D = 1] P(D = 1) + E[X1| D=2]P(D=2) = 1*1/6 + 1*1/6 = 1/3

So due to law of large numbers if you did this experiment N >> 1 times you would expect (~N/3) occurrences of 1 or 6. Letting X1 be 1 if the first die is 6 (0 otherwise) and X2 be 1 if the second die is 6 (0 otherwise)

E[X1 + X2] = E[X1] +E[X1] = 1*1/6 + 1*1/6 = 1/3

So again after N>>1 tries you would expect (~N/3) 6s.
 
  • #11
HallsofIvy said:
By the way, there is no such word (in English) as "dices". "Dice" is the plural of "die".

"Dice" is the plural of "die," as when referencing a gambling object,

but "dices" is a plural to "dice," as in "cutting up an apple into dices."

Also, "dices" is a verb.

And "die," when meaning a machine that stamps/cuts out a shape,
has the plurals "dies" and "dice."


http://www.answers.com/topic/dice


http://www.answers.com/topic/die-manufacturing
 
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  • #12
Thanks, guys, I bow to superior knowledge.
 
  • #13
Help me please. Maybe I'm misunderstanding the original question. As I read it he is asking "what's the probability of rolling a 1 or a 6 on one roll of a die?" Clearly 1/3. As I read the second question, he is asking "what is the probability of rolling at least one 6 on a roll of two dice?" The answer is the complement of getting no 6's on the roll of two dice, or 1-(5/6)(5/6)=11/36. You guys appear to be saying that they are equal. How am I misreading the question? Thanks.
 
  • #14
No he was asking how many times 6s would come up, so you count the 6 - 6 case as 2
 
  • #15
So he was asking for the expected number of 1's or 6's on a single roll of the die vs. the expected number of 6's in a roll of two dice? I didn't read that, thanks. Both expectations are 1/3.
 
  • #16
yup! ;)
 
  • #17
Thanks, it helps to be talking about the same problem.
 
  • #18
Ups sorry, noticed a typo in my other post should say:
kai_sikorski said:
E[X1 + X2] = E[X1] +E[X2] = 1*1/6 + 1*1/6 = 1/3
 

What is the probability of rolling a 6 or 1 with two dice?

The probability of rolling a 6 or 1 with two dice is 1/3 or approximately 33.33%. This can be calculated by dividing the number of outcomes that result in a 6 or 1 (11) by the total number of possible outcomes (36).

What is the concept of an infinite experiment?

An infinite experiment is one that has an infinite number of possible outcomes. In the case of rolling two dice, there are an infinite number of possible outcomes because the dice can be rolled an infinite number of times.

How does the probability of rolling a 6 or 1 change with each roll?

The probability of rolling a 6 or 1 remains constant with each roll. This is because each roll is an independent event and the outcome of the previous roll does not affect the outcome of the next roll. Therefore, the probability of rolling a 6 or 1 with two dice will always be 1/3, regardless of how many times the dice are rolled.

Is it possible to roll a 6 or 1 with two dice on every single roll?

While it is possible to roll a 6 or 1 with two dice on every single roll, it is highly unlikely. The probability of this occurring is 1/3 raised to the power of infinity, which is essentially 0. Therefore, it is theoretically possible but practically impossible to roll a 6 or 1 on every single roll.

How does the probability of rolling a 6 or 1 with two dice compare to other outcomes?

The probability of rolling a 6 or 1 with two dice is relatively high compared to other outcomes. Out of the 36 possible outcomes, 11 result in a 6 or 1. This means that the probability of rolling a 6 or 1 is higher than the probability of rolling any other specific number or combination of numbers. However, the probability of rolling a 6 or 1 is still relatively low compared to the probability of rolling any number between 2 and 12, which is 100%.

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