A red die, a blue die, and a yellow die (all six sided) are rolled. we are interested in the probability that the number appeari
ng on the blue die is less than that appearing on the yellow die, which is less than that appearing on the red die. that is, with b, y, and r denoting, respectively, the number appearing on the blue, yellow, and red die, we are interested in p(b < y < r).
5/54 or approximately 0.092592593 There are 6^3 = 216 possible outcomes of rolling these 3 dice. Let's count the number of possible rolls that meet the criteria b < y < r, manually. r = 1 or 2 is obviously impossible. So let's look at r = 3 through 6. r = 3, y = 2, b = 1 is the only possibility for r=3. So n = 1 r = 4, y = 3, b = {1,2}, so n = 1 + 2 = 3 r = 4, y = 2, b = 1, so n = 3 + 1 = 4 r = 5, y = 4, b = {1,2,3}, so n = 4 + 3 = 7 r = 5, y = 3, b = {1,2}, so n = 7 + 2 = 9 r = 5, y = 2, b = 1, so n = 9 + 1 = 10
And I see a pattern, for the most restrictive r, there is 1 possibility. For the next most restrictive, there's 2+1 = 3 possibilities. Then the next one is 3+2+1 = 6 possibilities. So for r = 6, there should be 4+3+2+1 = 10 possibilities. Let's see r = 6, y = 5, b = {4,3,2,1}, so n = 10 + 4 = 14 r = 6, y = 4, b = {3,2,1}, so n = 14 + 3 = 17 r = 6, y = 3, b = {2,1}, so n = 17 + 2 = 19 r = 6, y = 2, b = 1, so n = 19 + 1 = 20 And the pattern holds. So there are 20 possible rolls that meet the desired criteria out of 216 possible rolls. So 20/216 = 5/54.
1. In number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: Euclid's lemma — If a prime p divides the product ab of two integers a and b, then p must divide at least one of those integers a and b.
2. The Thales theorem states that: If three points A, B, and C lie on the circumference of a circle, whereby the line AC is the diameter of the circle, then the angle ∠ABC is a right angle (90°).
