 ## ________________

The coach decides to give the team a workout with a couple of probability problems.

## ________________ ### Fifty-Fifty

From a bag containing red and blue balls, two are removed at random. The chances are 50-50 that they will differ in color.

What were the possible numbers of balls initially in the bag?

### Random Toss

Teammate Joachim is tossing circular disks randomly on a floor marked out in squares. He wins if the disk lands so that it covers parts of exactly two or three squares. In the diagram, A and B are winning positions, while C and D are not. Suppose the squares have a unit side and Joachim gets to pick the diameter of the disk.

What probability of winning can Joachim achieve and what disk diameter achieves this? ## Solutions to week 116

Solutions for Divide by Three and Cutting the Domino are shown below.

Divide by Three Cutting the Domino Divide by Three answer explained:
Scale the half hexagon so that its three sides are of unit length. In the first solution, the smaller length is the cube root of 2 and the larger length is the cube root of 4. In the second solution, the three lengths are (9-√33)/12, (√33-3)/6, and (15-√33)/12. Cutting the Domino answer explained:
In (a1), the pieces are right triangles with legs in the ratio of 2 to 1.
In (a2), the pieces are right triangles with legs in the ratio of 1 to 1.
In (b), the ratio of sizes of the small pieces to the large piece is f = 2-√(3.5) = .12917…
The cut length is C = 5-6√(3.5) = .775028… ## Recent Weeks

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

Come back next week for answers and more puzzles.