Answer:
Chebyshev's inequality provides an upper bound on the probability that the total number of dots when a fair die is tossed 100 times falls between 300 and 400 as at least 75%. It is a conservative estimate, and the actual probability is likely higher due to the law of large numbers.
Explanation:
The Chebyshev's inequality can be used to provide an upper bound on the probability that the total number of dots when a fair die is tossed 100 times is between 300 and 400.
Chebyshev's Inequality states that for any random variable X (with finite mean μ and finite variance σ^2), the probability that X deviates from its mean by more than k standard deviations is at most 1/k^2, where k is a positive constant.
In this case, the random variable X represents the total number of dots when a fair die is tossed 100 times. The mean (μ) of X can be calculated as the mean of a single toss of a fair die times 100. The variance (σ^2) of X can be calculated as the variance of a single toss times 100 since each toss is independent.
Mean of a single toss of a fair die = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5
Mean of X = 100 * 3.5 = 350
Variance of a single toss of a fair die = [(1 - 3.5)^2 + (2 - 3.5)^2 + (3 - 3.5)^2 + (4 - 3.5)^2 + (5 - 3.5)^2 + (6 - 3.5)^2] / 6
Variance of a single toss = 35/12 ≈ 2.9167
Variance of X = 100 * 2.9167 ≈ 291.67
Now, let's calculate the standard deviation (σ) of X:
σ = √(Variance of X) = √291.67 ≈ 17.09
We want to find the probability that X deviates from its mean (350) by more than some value k * σ and falls between 300 and 400. Therefore, we need to calculate k.
Let's choose k = 2 (a common choice for Chebyshev's inequality). This means we are interested in the probability that X deviates from its mean by more than 2 standard deviations.
Now, we can use Chebyshev's inequality:
P(|X - μ| > kσ) ≤ 1/k^2
P(|X - 350| > 2 * 17.09) ≤ 1/2^2
P(|X - 350| > 34.18) ≤ 1/4
Now, let's calculate the probability that X is between 300 and 400:
P(300 ≤ X ≤ 400) = 1 - P(|X - 350| > 34.18) ≥ 1 - 1/4 = 3/4
So, using Chebyshev's inequality, we can bound the probability that the total number of dots is between 300 and 400 as at least 75%. This is a conservative bound, and the actual probability is likely higher due to the properties of the fair die and the law of large numbers, which suggests that as the number of tosses increases, the results converge to the expected mean more closely.