Comparison

Solution & Explanation

To solve the problem of determining which scenario has the highest probability, we can utilize the concept of binomial probability. The binomial probability formula is used to find the probability of achieving a certain number of successes in a fixed number of independent trials. The formula is:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

Where:

• $n$ is the number of trials,
• $k$ is the number of successes,
• $p$ is the probability of success in a single trial,
• $\binom{n}{k}$ is the binomial coefficient.

In this problem, the probability of rolling a six on a fair six-sided die is $p = \frac{1}{6}$, and the probability of not rolling a six is $1-p = \frac{5}{6}$.

Scenario 1: At least one six in 6 rolls

To find the probability of rolling at least one six in 6 rolls, we calculate the complement of rolling no sixes at all:

$P(X \ge 1) = 1 - P(X = 0)$

$P(X = 0) = \left(\frac{5}{6}\right)^6$

$P(X \ge 1) = 1 - \left(\frac{5}{6}\right)^6 \approx 0.6651$

Scenario 2: At least two sixes in 12 rolls

For at least two sixes in 12 rolls, we need to find the complement of having zero or one six:

$P(X \ge 2) = 1 - (P(X = 0) + P(X = 1))$

$P(X = 0) = \left(\frac{5}{6}\right)^{12}$

$P(X = 1) = \binom{12}{1} \left(\frac{1}{6}\right)^1 \left(\frac{5}{6}\right)^{11}$

$P(X \ge 2) = 1 - \left(\left(\frac{5}{6}\right)^{12} + 12 \cdot \left(\frac{1}{6}\right) \cdot \left(\frac{5}{6}\right)^{11}\right) \approx 0.6187$

Scenario 3: At least 100 sixes in 600 rolls

This scenario involves a large number of trials, making it appropriate to use a normal approximation to the binomial distribution:

• Mean $\mu = n \cdot p = 600 \cdot \frac{1}{6} = 100$
• Standard deviation $\sigma = \sqrt{n \cdot p \cdot (1-p)} = \sqrt{600 \cdot \frac{1}{6} \cdot \frac{5}{6}} \approx 8.16$

Using the normal approximation:

$P(X \ge 100) \approx P\left(Z \ge \frac{100 - 100}{8.16}\right) = P(Z \ge 0) = 0.5$

Conclusion

Comparing the probabilities:

• Probability of at least one six in 6 rolls: $0.6651$
• Probability of at least two sixes in 12 rolls: $0.6187$
• Probability of at least 100 sixes in 600 rolls: $0.5$

Thus, rolling at least one six in 6 rolls has the highest probability, making it the most likely scenario.