On February 2, 1894, Frank Raphael Weldon wrote a letter to Francis Galton that included the results of 26,306 rolls of 12 dice. Weldon recorded the results such that a roll of a 5 or 6 resulted in a success, while a roll of 1, 2, 3, or 4 was a failure. The number of successes in each roll of the 12 dice are recorded in the table.

(a) What is the probability of rolling a 5 or 6 when throwing a six-sided fair die?
(b) Treating the probability determined in part (a) as the probability of success, compute the theoretical probability of 0, 1, 2, ... , 12 successes in throwing 12 dice.
(c) Use the probabilities found in part (b) to determine the expected frequency in observing 0, 1, 2, ... , 12 successes after throwing the 12 dice 26,306 times.
(d) Conduct a goodness-of-fit test to determine if the number of successes follows a binomial probability distribution.

Number of Frequency Number of Frequency Successes Successes 185 1331 1149 403 105 3265 9. 5475 10 14 6114 11 5194 12 6. 3067 2. 3. 4)