The approximate probability that the student monitor calculated an average number of sixes per student greater than 2.2 is 0.9987.
Here's the solution to the problem about the die rolls:
Approximate Probability of High Mean Sixes:
We can approach this problem using statistical methods, specifically by considering the binomial distribution and the central limit theorem.
Here's the step-by-step solution:
Binomial Distribution: Each die roll is a Bernoulli trial with probability of success (rolling a six) being 1/6 and probability of failure (not rolling a six) being 5/6.
The number of sixes in 12 throws follows a binomial distribution with parameters n = 12 (number of trials) and p = 1/6 (probability of success).
Central Limit Theorem:
As we have 40 independent trials (students throwing dice), the central limit theorem tells us that the average number of sixes across all students will approximately follow a normal distribution, even if the individual trials follow a binomial distribution.
This is true even if the sample size (40) is not very large.
Calculating the Probability: We are interested in the probability that the average number of sixes is greater than 2.2.
We can use the normal distribution with mean (expected average) equal to np (number of trials) p (probability of success) = 12 * (1/6) = 2 and standard deviation equal to sqrt(np * p * (1-p)) = sqrt(12 * (1/6) * (5/6)) ≈ 1.29.
Using Statistical Tools: Using statistical software libraries like scipy.stats in Python, we can calculate the probability that a normally distributed variable with mean 2 and standard deviation 1.29 is greater than 2.2.
This probability is approximately 0.9987.
Therefore, the approximate probability that the student monitor calculated an average number of sixes per student greater than 2.2 is 0.9987.
This is a very high probability, suggesting that it is quite likely that the average number of sixes will be above 2.2 in this scenario.