Question

A box contains 24 tiles. Each tile has certain properties:

Each tile is made of wood or plastic.
Each tile is red, yellow, or blue.
Each tile is a triangle, square, pentagon, or circle.

There is exactly one tile for each combination of properties. For example, there is one red circle that is made of wood.

Steve chooses one tile, and Ellen chooses a different tile. What is the probability that the two chosen tiles have exactly two properties in common?

Answer 1

There are 2 choices for the material of the first tile, 3 choices for the color, and 4 choices for the shape. This gives $2\cdot 3\cdot 4=24$ choices for the first tile. After one tile has been chosen, there are 23 choices for the second tile. The number of successful choices for the second tile is 4, since the second tile must have the same color and the same shape as the first tile, but a different material. Therefore, the probability that the two tiles have exactly two properties in common is $\frac{4}{23}$.

Alternatively, since the probability of choosing any given tile is $\frac{1}{24}$, the probability that the two chosen tiles are the same is $\left(\frac{1}{24}\right)\left(\frac{1}{23}\right)$.

However, this probability is not what we are looking for. We seek to find the probability that the two chosen tiles have exactly two properties in common. There are 2 choices for the different property, so this probability is $1-\left(\frac{1}{24}\right)\left(\frac{1}{23}\right)\left(2\right)=\boxed{\frac{11}{46}}$.