Three dice are thrown simultaneously. What is the probability that the sum on the three faces is at least 5?

• A: 17/18
• B: 53/54
• C: 103/108
• D: 215/216

To find the probability that the sum of the numbers on three thrown dice is at least 5, we first need to determine the total number of possible outcomes when rolling three dice.

Total Outcomes

Each die has 6 faces, so when rolling three dice, the total number of outcomes is:

• 6 (for the first die) × 6 (for the second die) × 6 (for the third die) = 216 total outcomes.

Favorable Outcomes

Next, we need to find the number of outcomes where the sum is less than 5. The only possible sums that are less than 5 when rolling three dice are 3 and 4.

Sum of 3

The only combination to achieve a sum of 3 is:

• (1, 1, 1)

This gives us 1 outcome.

Sum of 4

The combinations to achieve a sum of 4 are:

• (1, 1, 2)
• (1, 2, 1)
• (2, 1, 1)

There are 3 outcomes for this sum.

Calculating Total Outcomes Less Than 5

Adding these outcomes together, we have:

• 1 (for sum of 3) + 3 (for sum of 4) = 4 outcomes.

Finding the Probability

The number of favorable outcomes for the sum being at least 5 is:

• Total outcomes - Outcomes less than 5 = 216 - 4 = 212 outcomes.

The probability of the sum being at least 5 is then:

• Probability = Favorable outcomes / Total outcomes = 212 / 216.

This simplifies to:

• 53 / 54.

Final Answer

The probability that the sum on the three faces is at least 5 is B: 53/54.