To solve this problem, let us first calculate the probability of designing one shoe out of all the combinations.
Probability of designing one shoe = (probability of choosing 1 color for the base) * (probability of choosing 1 color for the accent) * (probability of choosing 1 font for the embroidery)
We know that probability is:
Probability = number of chosen event / number of total events
Therefore:
Probability of designing one shoe = (1 / 4) * (1 / 6) * (1 / 2)
Probability of designing one shoe = 1 / 48
This means that either Ray or Lilli has this probability of designing a specific shoe. Since we are asked for the probability that they will both design this specific shoe, so we multiply:
Probability they will design the same shoe = (1 / 48) * (1 / 48)
Probability they will design the same shoe = 1 / 2304 = 4.34 * 10^-4
The probability is around 0.043% that they will design the same shoe.