47. (Newton–Pepys problem) In 1693, Samuel Pepys, an English Naval administrator and member of the Parliament, corresponded with Isaac Newton concerning the following wager he intended to make: Which of the following trials is more likely than the other two?

(a) At least one 6 when 6 dice are tossed.

(b) At least two 6’s when 12 dice are tossed.

(c) At least three 6’s when 18 dice are tossed.

Isaac Newton calculated the probabilities of these events correctly and provided Samuel Pepys with correct answers. He also justified his answers intuitively; however, his justifications were invalid. Calculate the probability of the events above posed by Samuel Pepys.