A fair die is rolled repeatedly. Use precise notation of probabilities of events and random variables for the solutions to the questions below.

(a) Write down a precise sum expression for the probability that the first five rolls give a 3 at most two times.

(b) Calculate the probability that the first 3 does not appear before the fifth roll.

(c) Calculate the probability that the first 3 appears before the twentieth roll but not before the fifth roll

The number of nails of a given length is normally distributed with a mean length of 5.00 in, and a standard deviation of 0.03 in. find the number of nails in a bag of 120 that are less than 4.94 in. long.

Points a, b, and c form a triangle. complete the statements to prove that the sum of the interior angles of triangle abc is 180

1- what do you think the product of a nonzero rational number and an irrational number is? is it rational or irrational? make use of variables, the closure property of integers, and possibly a proof by contradiction to prove your hypothesis.2- why do we have to specify that the rational number must be nonzero when we determine what the product of a nonzero rational number and an irrational number is? if the rational number were 0, would it give us the same result we found in the first question?