Three dice are thrown together. find the probability of getting a total of at least 6?

General
1

three dice are thrown together. find the probability of getting a total of at least 6 ?

2

Three dice are thrown together. Find the probability of getting a total of at least 6.

Answer:

When three dice are thrown together, the total number of possible outcomes is 6^3 = 216 because each die has 6 faces. Now, to find the probability of getting a total of at least 6, we must first identify all the favorable outcomes.

Let’s list the combinations where the total of the three dice is at least 6:

  • (2, 2, 2) - 3 ways
  • (2, 2, 3), (2, 3, 2), (3, 2, 2) - 6 ways
  • (1, 2, 4), (1, 4, 2), (2, 1, 4), (2, 4, 1), (4, 1, 2), (4, 2, 1), (3, 3, 3) - 12 ways
  • (1, 2, 5), (1, 5, 2), (2, 1, 5), (2, 5, 1), (5, 1, 2), (5, 2, 1), (3, 3, 4), (3, 4, 3), (4, 3, 3) - 18 ways
  • (1, 2, 6), (1, 6, 2), (2, 1, 6), (2, 6, 1), (6, 1, 2), (6, 2, 1), (3, 3, 5), (3, 5, 3), (5, 3, 3), (4, 4, 4) - 21 ways
  • (1, 3, 6), (1, 6, 3), (3, 1, 6), (3, 6, 1), (6, 1, 3), (6, 3, 1), (2, 4, 6), (2, 6, 4), (4, 2, 6), (4, 6, 2), (6, 2, 4), (6, 4, 2), (5, 5, 5) - 24 ways
  • (1, 4, 6), (1, 6, 4), (4, 1, 6), (4, 6, 1), (6, 1, 4), (6, 4, 1), (2, 5, 6), (2, 6, 5), (5, 2, 6), (5, 6, 2), (6, 2, 5), (6, 5, 2), (3, 4, 6), (3, 6, 4), (4, 3, 6), (4, 6, 3), (6, 3, 4), (6, 4, 3) - 27 ways
  • (1, 5, 6), (1, 6, 5), (5, 1, 6), (5, 6, 1), (6, 1, 5), (6, 5, 1), (2, 3, 6), (2, 6, 3), (3, 2, 6), (3, 6, 2), (6, 2, 3), (6, 3, 2), (4, 5, 6), (4, 6, 5), (5, 4, 6), (5, 6, 4), (6, 4, 5), (6, 5, 4) - 30 ways
  • (1, 3, 5), (1, 5, 3), (3, 1, 5), (3, 5, 1), (5, 1, 3), (5, 3, 1), (2, 3, 5), (2, 5, 3), (3, 2, 5), (3, 5, 2), (5, 2, 3), (5, 3, 2) - 30 ways

So, the total number of favorable outcomes where the sum is at least 6 is 171.

Therefore, the probability of getting a total of at least 6 when throwing three dice together is \frac{171}{216} = \frac{19}{24} .