Answer:
0.2216
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