Question

once lucia finishes placing the tiles in her art project, she needs to determine the color of the grout that goes in between the tiles and the color of the frame around the project. she flips a coin to decide if she is going to use blue or yellow grout. she assigns heads to blue grout and tails to yellow grout. she puts a yellow, green, blue, and purple tile in a bag and pulls one out to determine the frame color.
a. determine the possible outcomes for flipping a coin once and selecting one tile out of the bag at random.
b. how many possible outcomes are there?
c. what events make up the outcome of having the same color for grout and the frame at random?
d. determine the probability of the outcome where the same color for grout and the frame are chosen at random.

Explanation:

Step1: List valid combined outcomes

Coin (grout): Heads=Blue (B), Tails=Yellow (Y)
Tiles (frame): Yellow (Y), Green (G), Blue (B), Purple (P)
Combined outcomes: $(B,Y), (B,G), (B,B), (B,P), (Y,Y), (Y,G), (Y,B), (Y,P)$

Step2: Count total outcomes

Count all unique combined pairs:
$n(\text{total}) = 2 \times 4 = 8$

Step3: Identify matching color outcomes

Find pairs where grout and frame color match:
$(B,B), (Y,Y)$

Step4: Calculate matching outcome count

Count the matching pairs:
$n(\text{matching}) = 2$

Step5: Compute probability

Probability = $\frac{\text{Matching outcomes}}{\text{Total outcomes}}$
$\text{Probability} = \frac{2}{8} = \frac{1}{4}$

Answer:

a. The possible outcomes are: $(B,Y), (B,G), (B,B), (B,P), (Y,Y), (Y,G), (Y,B), (Y,P)$
b. 8
c. The events are (Blue grout, Blue frame) and (Yellow grout, Yellow frame), or written as $(B,B)$ and $(Y,Y)$
d. $\frac{1}{4}$