# Calculating Exact Probability of Rolling Two Dice

• kuahji
In summary, the conversation discusses the use of Chebyshev's Theorem to find the lower bound for the probability that a random variable X, representing the sum of two fair dice rolls, is within two standard deviations of the mean. The mean and variance are calculated to be 7 and 35/6 respectively. Part A of the question is solved to find the lower bound to be 3/4. Part B asks for the exact probability, which is found to be 11/36 using the probability mass function of X. The conversation concludes with a request for clarification on the answer given by the professor in class.
kuahji
Hoping this will be my last stats question for awhile.

X is the random variable for the sum obtained by rolling two fair dice.

I figured out the mean & variance which are 7 & 35/6.

But I then get the question part A) "Use Chebyshev's Theorem to find the lower bound for the probability that a value X is within two standard deviations of the mean X." Which I computed to be 3/4.
Part B) of that question says "Determine the exact probability that this sum is within the same two standard deviations."
Here I'm kinda lost. I recall in class, I think the answer was 34/36, but the professor didn't elaborate.

Any help would be appreciated.Thanks!The exact probability that X is within two standard deviations of the mean is 11/36. This can be calculated using the probability mass function of the distribution of X. The probability mass function gives the probability of obtaining each value of X (in this case, 2 to 12). If you add together the probabilities for all values of X between 6 and 8 (including 6 and 8), you get 11/36.

## What is the probability of rolling a specific number with two dice?

The probability of rolling a specific number (such as a 7) with two dice is 1/6, or approximately 16.67%. This is because there are 6 possible outcomes for each die (1, 2, 3, 4, 5, 6), and when rolling two dice, the outcomes are multiplied together to get the total number of possible combinations (6x6=36). Out of those 36 possible combinations, only one will result in a specific number (such as 7).

## What is the probability of rolling a certain sum with two dice?

The probability of rolling a certain sum (such as a sum of 8) with two dice can be calculated by finding the number of ways to get that sum and dividing it by the total number of possible combinations (36). For example, there are 5 ways to get a sum of 8 with two dice (2+6, 3+5, 4+4, 5+3, 6+2), so the probability would be 5/36, or approximately 13.89%.

## What is the probability of rolling a double with two dice?

The probability of rolling a double (both dice showing the same number) with two dice is 1/6, or approximately 16.67%. This is because there are 6 possible outcomes for each die, and when rolling two dice, the outcomes are multiplied together to get the total number of possible combinations (6x6=36). Out of those 36 possible combinations, only 6 will result in a double (1+1, 2+2, 3+3, 4+4, 5+5, 6+6).

## What is the probability of rolling a total less than a certain number with two dice?

The probability of rolling a total less than a certain number (such as less than 4) with two dice can be calculated by finding the number of combinations that result in a total less than that number and dividing it by the total number of possible combinations (36). For example, there are 3 combinations that result in a total less than 4 (1+1, 1+2, 2+1), so the probability would be 3/36, or approximately 8.33%.

## How can I calculate the exact probability of rolling two dice?

The exact probability of rolling two dice can be calculated by dividing the number of desired outcomes by the total number of possible outcomes. For example, if you want to know the probability of rolling a sum of 9, you would count the number of combinations that result in a sum of 9 (4 out of 36) and divide it by the total number of combinations (36), giving you a probability of 4/36, or approximately 11.11%. This method can be used for any desired outcome or combination of outcomes.

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