A title is randomly selected from a bag that contains 10 tiles . The tales either blue or, green , red or yellow . After the color is recorded the tiles is placed back in a bag , a tiles is pulled 100 times of which 24 are blue tiles 12 are green tiles and 12 are red tiles . How many yellow tiles are most likely in the bag .?

Question

GPT-4o mini · Answer

Let's define the variables based on the information provided:

Let \( b \) be the number of blue tiles.
Let \( g \) be the number of green tiles.
Let \( r \) be the number of red tiles.
Let \( y \) be the number of yellow tiles.

From the problem, we know the following:

The total number of tiles is 10: \[ b + g + r + y = 10 \]
Over 100 draws, the number of tiles drawn is:
- 24 blue tiles
- 12 green tiles
- 12 red tiles
- And the rest are yellow tiles.

Given that the tiles are drawn 100 times, the total counts for blue, green, and red are: \[ \text{Total drawn} = 100 \] \[ \text{Yellow tiles drawn} = 100 - (24 + 12 + 12) = 100 - 48 = 52 \]

The observed frequency of draws indicates:

The proportion of blue tiles drawn is \( \frac{24}{100} = 0.24 \)
The proportion of green tiles drawn is \( \frac{12}{100} = 0.12 \)
The proportion of red tiles drawn is \( \frac{12}{100} = 0.12 \)
The proportion of yellow tiles drawn is \( \frac{52}{100} = 0.52 \)

Assuming the draws represent the actual proportions of the tiles in the bag, we can establish the expected number of each color tile in terms of the total number of tiles, which is 10.

Thus, we compute:

Expected blue tiles: \( 0.24 \cdot 10 = 2.4 \)
Expected green tiles: \( 0.12 \cdot 10 = 1.2 \)
Expected red tiles: \( 0.12 \cdot 10 = 1.2 \)
Expected yellow tiles: \( 0.52 \cdot 10 = 5.2 \)

Since the number of tiles must be whole numbers, let's round these to the nearest whole numbers.

For practical purposes, we can deduce:

Blue tiles (\( b = 2 \))
Green tiles (\( g = 1 \))
Red tiles (\( r = 1 \))
Yellow tiles (\( y = 5 \))

Now, let's check our result against the total of 10: \[ b + g + r + y = 2 + 1 + 1 + 5 = 9 \]

Since this does not equal 10, we can adjust our approach slightly. The closest integers maintaining the ratios while ensuring the total is 10 means increasing the blue or any other color slightly or adjusting yellow downwards.

Let’s try:

Blue = 2
Green = 1
Red = 1
Yellow = 6 (since we have 2 blue, 1 green, 1 red counted, the total is now even more plausible).

This totals correctly as: \[ 2 + 1 + 1 + 6 = 10 \]

Thus, the most likely number of yellow tiles in the bag is 6.