A die with six faces is rolled 900 times. Each of the six faces is equally likely to come up
The probability that the sum is between 3099 and 3252 is .
not sure!
To find the probability that the sum of the rolls is between 3099 and 3252, we need to consider the different combinations of rolls that satisfy this condition.
Each roll of the die has a face value ranging from 1 to 6. Assuming the rolls are independent of each other and equally likely, we can use the concept of binomial distribution to approximate the probability.
Let's break it down step by step:
1. Determine the range of possible sums:
- The minimum sum can be obtained by rolling a 1 on every single roll: 1 * 900 = 900.
- The maximum sum can be obtained by rolling a 6 on every single roll: 6 * 900 = 5400.
So, the possible sum range is between 900 and 5400.
2. Calculate the mean and standard deviation of a single roll:
- The mean of a single roll is given by (1+2+3+4+5+6)/6 = 3.5.
- The standard deviation of a single roll is given by sqrt(((1-3.5)^2 + (2-3.5)^2 + ... + (6-3.5)^2)/6) ≈ 1.71.
3. Calculate the mean and standard deviation of the sum of 900 rolls:
- The mean of the sum of 900 rolls is given by 900 * 3.5 = 3150.
- The standard deviation of the sum of 900 rolls is given by sqrt(900) * 1.71 ≈ 46.54.
4. Standardize the range of sums:
- Subtract the mean of the sum from each boundary to get a new range: (3099 - 3150)/(46.54) to (3252 - 3150)/(46.54).
- This gives us the standardized range from -2.17 to 2.18.
5. Use a standard normal distribution table or calculator to find the probability:
- Look up the probability for the range -2.17 to 2.18 under the standard normal distribution curve.
- Alternatively, you can use a calculator or software to calculate the probability directly using the z-scores.
Note: Since the number of rolls is large (900), we can use the approximation that the sum of rolls follows a normal distribution due to the central limit theorem.