Roll a fair die 6 times and read the number on the upper face. (a) what’s the probability of observing all possible numbers 1 to 6? (b) what’s the probability of observing three 6-s and three 1-s? (c) what’s the probability of observing two 1-s, two 3-s and two 5-s? (d) what’s the probability of observing six even numbers?

a) every number is a 1/6 chance so 6×1/6 = 1/36

b) also 1/36 because the chance doesnt change. if you throw 6 twice, the chance for a third time is still 1/6.

c) 1/36

d) 2, 4 and 6 are even so now the chance is 3/6 = 1/2 and that times 6

4x and 6y

im not sure though

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.