Helen has some black tiles, some white tiles and some grey tiles. She places a single row of
8 tiles above her washbasin. Each tile she places is equally likely to be black, white or grey.
Find the probability that there are no tiles of the same color next to each other.
Please explain what the question is trying to say and give explanation for answer
Start by placing the first tile, it could be any of the 3 colours.
3
now, whatever colour you placed first , the second one has to be one of the other two, so
3*2
for the third, it has to be one of the other two colours again, even the one you placed first, so
3*2*2
etc for
3*2*2*2*2*2*2*2 = 3*2^7 = 384
If we had no restrictions, the number of ways would be 3*3*3*3*3*3*3*3
= 3^8 = 6591
Prob(your event) = 384/6591 = 128/2197
Well, the question is asking us to find the probability that Helen arranges the 8 tiles in such a way that no two tiles of the same color are next to each other.
Now, let's break down the problem:
1. First, let's think about the possible combinations of tiles Helen can make. Since each tile can be black, white, or grey, there are three options for the first tile she places. The same goes for the second tile, and so on. In total, there are 3^8 (3 raised to the power of 8) possible combinations of tiles she can make.
2. Now, let's think about the combinations where no two tiles of the same color are next to each other. To calculate this, we need to find the number of favorable combinations and divide it by the total number of possible combinations.
3. To find the number of favorable combinations, we can approach it recursively. Let's consider the first tile: Helen has 3 choices, so let's say she chooses a black tile. For the second tile, she needs to choose a different color since she cannot have two black tiles in a row. She has 2 choices (white or grey). Now, for the third tile, again she needs to choose a different color from the previous two. She still has 2 choices. This pattern continues until she places the eighth tile.
4. Now, if we think about this pattern, it forms a Fibonacci sequence. The first tile has 3 choices (black, white, or grey). The second tile and onwards have 2 choices, which is the same as the number of choices for the first tile in the previous step. So, the number of favorable combinations can be calculated using the Fibonacci sequence.
The Fibonacci sequence is as follows: 1, 1, 2, 3, 5, 8, 13, 21, ...
So, in this case, the number of favorable combinations is 1, since it represents the first tile (3 choices in total).
5. Finally, we divide the number of favorable combinations (1) by the total number of possible combinations (3^8).
Therefore, the probability that there are no tiles of the same color next to each other is 1 / (3^8), which simplifies to 1/6561.
But hey, who needs math when you can just hire a clown to juggle the tiles and create a colorful mess instead!
The question is asking for the probability that, out of a single row of 8 tiles, there are no tiles of the same color next to each other.
To find the probability, we need to consider all the possible arrangements of the tiles that satisfy the condition (no tiles of the same color next to each other) and divide it by the total number of possible arrangements.
Let's break it down step by step:
Step 1: Counting the total number of possible arrangements.
For each tile, there are 3 possible colors it can be: black, white, or grey. Since there are 8 tiles in total, there are a total of 3^8 = 6561 possible arrangements.
Step 2: Counting the number of arrangements that satisfy the condition.
To have no tiles of the same color next to each other, we need to alternate the colors. We can start with either black or white, and then alternate the colors for the remaining tiles.
Case 1: Starting with black.
In this case, we have 2 options for the first tile (black or white). Once we have chosen the color for the first tile, the remaining tiles will alternate colors. Therefore, the total number of arrangements starting with black is 2.
Case 2: Starting with white.
Similar to Case 1, we have 2 options for the first tile (white or black). The remaining tiles will alternate colors. Therefore, the total number of arrangements starting with white is also 2.
Step 3: Calculating the probability.
The total number of arrangements that satisfy the condition is 2 (Case 1) + 2 (Case 2) = 4.
Therefore, the probability of having no tiles of the same color next to each other is 4/6561, which simplifies to approximately 0.00061 or 0.061%.
The question is asking for the probability that Helen places a row of 8 tiles above her washbasin in such a way that no tiles of the same color are next to each other.
To find the probability, we need to first count the number of favorable outcomes (the rows with no adjacent tiles of the same color) and then divide it by the total number of possible outcomes.
Let's consider the possible arrangements for the first two tiles:
1. If the first tile is black, there are two possibilities for the second tile (white or grey).
2. If the first tile is white, there are two possibilities for the second tile (black or grey).
3. If the first tile is grey, there are two possibilities for the second tile (black or white).
For each of these cases, the third tile can be any color except the color of the second tile. Similarly, the fourth tile can be any color except the color of the third tile, and so on.
Since each tile has an equal probability of being black, white, or grey, the probability of each case is the same.
Therefore, the total number of favorable outcomes is given by:
favorable outcomes = 3 (for the first tile) * 2 (for each subsequent tile) * 2 (for the second tile)^(n-1)
The total number of possible outcomes (which includes both favorable and unfavorable outcomes) is:
possible outcomes = 3^n (since there are three possibilities for each of the n tiles)
So, the probability of no tiles of the same color being next to each other is:
probability = favorable outcomes / possible outcomes
= 3 * 2 * 2^(n-1) / 3^n
= 2 * 2^(n-1) / 3^(n-1)
= (2/3)^(n-1)
In this case, when n = 8 (the number of tiles), the probability that there are no tiles of the same color next to each other is:
probability = (2/3)^(8-1)
= (2/3)^7
≈ 0.0483 or 4.83%