Ihave two bags. bag 1 contains 1010 blue marbles, while bag 2 contains 1515 blue marbles. i pick one of the bags at random, and throw 66 red marbles in it. then i shake the bag and choose 55 marbles (without replacement) at random from the bag. if there are exactly 22 red marbles among the 55 chosen marbles, what is the probability that i have chosen bag 1

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.