A peculiar six-sided die has uneven faces. In particular, the faces showing 1 or 6 are 1 * 1.5 inches, the faces showing 2 or 5 are 1 * 0.4 inches, and the faces showing 3 or 4 are 0.4 * 1.5 inches. Assume that the probability of a particular face coming up is proportional to its area. We independently roll the die twice. What is the probability that we get doubles?

0.2216

Step-by-step explanation:

The probability of a particular face coming up is proportional to its area.

Hence, we first calculate the areas.

For 1 or 6, area = 1 * 1.5 = 1.5 in²

For 2 or 5, area = 1 * 0.4 = 0.4 in²

For 3 or 4, area = 0.4 * 1.5 = 0.6 in²

Total surface area of the die = 2 (1.5) + 2 (0.4) + 2 (0.6) = 5 in²

Since probability is proportional to area of a particular face, we calculate the probabilities from the ratio of each surface area to the total surface area.

Probability of a 1 showing up = 1.5/5 = 0.30

Probability of a 2 showing up = 0.4/5 = 0.08

Probability of a 3 showing up = 0.6/5 = 0.12

Probability of a 4 showing up = 0.6/5 = 0.12

Probability of a 5 showing up = 0.4/5 = 0.08

Probability of a 6 showing up = 1.5/5 = 0.30

Probability of doubles with the die is a sum of the probabilities of all the possible doubles with the die.

Probability of a 1 double = 0.3 * 0.3 = 0.09

Probability of a 2 double = 0.08 * 0.08 = 0.0064

Probability of a 3 double = 0.12 * 0.12 = 0.0144

Probability of a 4 double = 0.12 * 0.12 = 0.0144

Probability of a 5 double = 0.08 * 0.08 = 0.0064

Probability of a 6 double = 0.3 * 0.3 = 0.09

Probability of a double with the die = (0.09 + 0.0064 + 0.0144 + 0.0144 + 0.0064 + 0.09) = 0.2216