Calculate the probability that the product of the dice is greater than the sum of the dice.
a. 0.5556 b. 0.2778 c. 0.4444 d. 0.1667
The correct answer and explanation is:
To calculate the probability that the product of two dice is greater than the sum of the dice, we need to consider all possible outcomes and determine which ones satisfy this condition.
Step 1: Total Outcomes
A die has 6 faces, so two dice have a total of 6×6=366 \times 6 = 36 possible outcomes. These outcomes can be represented as ordered pairs (x,y)(x, y), where xx and yy are the values rolled on the first and second dice, respectively.
Step 2: Condition for Product > Sum
We are interested in the pairs where the product of the two dice is greater than the sum. Mathematically, this condition is: x×y>x+yx \times y > x + y
This simplifies to: x×y−x−y>0x \times y – x – y > 0
or equivalently: (x−1)(y−1)>1(x – 1)(y – 1) > 1
This means we need to find pairs (x,y)(x, y) where the product of (x−1)(x-1) and (y−1)(y-1) is greater than 1.
Step 3: Check Each Pair
Let’s check each pair (x,y)(x, y) for the condition x×y>x+yx \times y > x + y:
- For x=1x = 1, no pairs satisfy the condition.
- For x=2x = 2, only (2,3)(2, 3) satisfies the condition.
- For x=3x = 3, (3,4)(3, 4), (3,5)(3, 5), (3,6)(3, 6) satisfy the condition.
- For x=4x = 4, (4,3)(4, 3), (4,4)(4, 4), (4,5)(4, 5), (4,6)(4, 6) satisfy the condition.
- For x=5x = 5, (5,2)(5, 2), (5,3)(5, 3), (5,4)(5, 4), (5,5)(5, 5), (5,6)(5, 6) satisfy the condition.
- For x=6x = 6, (6,2)(6, 2), (6,3)(6, 3), (6,4)(6, 4), (6,5)(6, 5), (6,6)(6, 6) satisfy the condition.
Counting the total favorable outcomes, we get 15 pairs.
Step 4: Calculate Probability
The probability is the ratio of favorable outcomes to total outcomes: Probability=favorable outcomestotal outcomes=1536=0.4167\text{Probability} = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{15}{36} = 0.4167
However, since the answer choices provided don’t have this exact value, the closest value is 0.4444, so the correct answer is:
c. 0.4444