To determine the probability that Gavin will select two red tiles (one from each bag), we need to follow a series of steps:
1. Calculate the total number of tiles in each bag:
- Bag 1: The total number of tiles in Bag 1 is the sum of all tiles in Bag 1.
[tex]\[
\text{Total Bag 1} = 4 \text{ (black)} + 6 \text{ (white)} + 3 \text{ (red)} + 5 \text{ (yellow)} = 18
\][/tex]
- Bag 2: The total number of tiles in Bag 2 is the sum of all tiles in Bag 2.
[tex]\[
\text{Total Bag 2} = 1 \text{ (black)} + 4 \text{ (white)} + 3 \text{ (red)} + 1 \text{ (yellow)} = 9
\][/tex]
2. Determine the probability of drawing a red tile from each bag:
- Probability of drawing a red tile from Bag 1:
[tex]\[
\text{Prob (Red Bag 1)} = \frac{\text{Number of red tiles in Bag 1}}{\text{Total number of tiles in Bag 1}} = \frac{3}{18} = \frac{1}{6} \approx 0.1667
\][/tex]
- Probability of drawing a red tile from Bag 2:
[tex]\[
\text{Prob (Red Bag 2)} = \frac{\text{Number of red tiles in Bag 2}}{\text{Total number of tiles in Bag 2}} = \frac{3}{9} = \frac{1}{3} \approx 0.3333
\][/tex]
3. Calculate the combined probability of both events happening:
- Since drawing one red tile from each bag are independent events, we multiply the probabilities:
[tex]\[
\text{Prob (Two Red Tiles)} = \text{Prob (Red Bag 1)} \times \text{Prob (Red Bag 2)} = \frac{1}{6} \times \frac{1}{3} = \frac{1}{18} \approx 0.0556
\][/tex]
Thus, the probability that Gavin will select two red tiles (one from each bag) is:
[tex]\[
\boxed{\frac{1}{18}}
\][/tex]
