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?
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 $1-\left(\frac{1}{24}\right)\left(\frac{1}{23}\right)\left(2\right)=\boxed{\frac{11}{46}}$.
To find the probability that the two chosen tiles have exactly two properties in common, we need to count the number of tiles that satisfy this condition and then divide it by the total number of possible combinations of tiles.
Step 1: Count the number of tiles that have exactly two properties in common.
Let's consider the three properties separately:
1) Material (wood or plastic): Since Steve and Ellen chose different tiles, the material property will always be different. So, there are no tiles that have the material property in common.
2) Color (red, yellow, or blue): Each color has 8 tiles (24 total tiles divided by 3 colors). Both Steve and Ellen can choose any of the 8 tiles of the same color, resulting in a total of 8 * 8 = 64 combinations.
3) Shape (triangle, square, pentagon, or circle): Each shape has 6 tiles (24 total tiles divided by 4 shapes). Again, both Steve and Ellen can choose any of the 6 tiles of the same shape, resulting in a total of 6 * 6 = 36 combinations.
The total number of tiles that have exactly two properties in common is 64 (from color) multiplied by 36 (from shape), which equals 2,304.
Step 2: Count the total number of possible combinations of tiles.
Since Steve and Ellen choose different tiles, each of them has 24 options to choose from. So the total number of possible combinations is 24 * 24 = 576.
Step 3: Calculate the probability.
Now divide the number of tiles that have exactly two properties in common (2,304) by the total number of possible combinations (576):
Probability = 2,304 ÷ 576 = 4
So, the probability that Steve and Ellen will choose tiles that have exactly two properties in common is 4 or 4/1, which can also be expressed as a probability of 1 in 4.
To calculate the probability that the two chosen tiles have exactly two properties in common, we need to determine the number of possible combinations where the two tiles share exactly two properties and then divide it by the total number of possible combinations.
Step 1: Calculate the number of possible combinations where the two tiles share exactly two properties:
- Properties can be shared in three different ways: two common properties (AA), one common property and one different property (AB), or two different properties (BB).
- Let's calculate each case:
1. Two common properties (AA):
- There are three properties to choose from (material, color, and shape).
- We can choose two properties out of three in (3 choose 2) = 3 ways.
- For each combination of two properties, there are 2 remaining properties that are not in common.
- The number of tiles with the chosen two common properties is 1.
Therefore, there are 3 * 2 * 1 = 6 combinations for the AA case.
2. One common property and one different property (AB):
- There are three properties to choose from.
- We can choose one property out of three in (3 choose 1) = 3 ways.
- For each combination of one common property, there is one property that is not in common.
- We now have two properties left to choose from, and we can choose one out of two in (2 choose 1) = 2 ways.
- The number of tiles with one common property and one different property is: 3 * 1 * 2 = 6 combinations.
3. Two different properties (BB):
- There are three properties to choose from.
- We can choose two properties out of three in (3 choose 2) = 3 ways.
- For each combination of two different properties, there is one property that is not in common.
- The number of tiles with two different properties is 3 * 2 * 1 = 6 combinations.
Summing up the combinations from each case: 6 + 6 + 6 = 18 combinations where the two tiles share exactly two properties.
Step 2: Calculate the total number of possible combinations:
- Each tile has 24 options.
- Steve chooses one tile, leaving 23 options remaining.
- Ellen chooses a different tile, leaving 22 options remaining.
- The total number of possible combinations is 24 * 23 = 552 combinations.
Step 3: Calculate the probability:
- The probability is the number of desired outcomes (combinations with exactly two properties in common) divided by the total number of outcomes (total combinations).
- The probability is: 18 / 552 ≈ 0.0326
Therefore, the probability that Steve and Ellen choose two tiles with exactly two properties in common is approximately 0.0326, or 3.26%.