## ________________

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?*

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## 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

**Week 116**: Cutting the Domino & Divide by Three, solutions to Thanksgiving Split & Easy as Pie

**Week 115**: Thanksgiving Split & Easy as Pie, solutions to Primes and Products & Two Pints of Cider

**Week 114**: Primes and Products & Two Pints of Cider, solutions to Prime Presents 1 & Prime Presents 2

**Week 113**: Prime Presents 1 & Prime Presents 2, solutions to Precise Prescription & Gift Dilemma

**Week 112**: Precise Prescription & Gift Dilemma, solutions to Pumpkin Crop & The Halloween Goblin

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

**Come back next week** for answers and more puzzles.