# Ideas in Geometry/Instructive examples/Lesson 21: Wacky Dice

Lesson 21: Wacky Dice

"OR" Lemma: If two events E1 and E2 cannot occur at the same time, then the probability that E1 or E2 will occur is the sum of the probabilities for each of the events
Probability (E1 or E2) = Probability(E1) + Probability(E2)

"AND" Lemma: If the occurrence of one of the events E1 and E2 does not affect the occurrence of the other, then the probability that E1 and E2 will occur is the product of the probabilities for each of the events
Probability (E1 and E2) = Probability(E1) x Probability(E2)

Using these two lemmas we can determine how often wacky die will win
We can see this by looking at the following example:
Die 1: 2, 2, 2, 2, 6, 6
Die 2: 1, 1, 1, 5, 5, 5

What is the probability Die 1 rolls a 2? 4/6 or 2/3
What is the probability Die 1 rolls a 6? 2/6 or 1/3
What is the probability Die 2 rolls a 1? 3/6 or 1/2
What is the probability Die 2 rolls a 5? 3/6 or 1/2

What is the probability that Die 1 rolls a 2 and Die 2 rolls a 1? Use the AND lemmato multiply probabilities
2/3 x 1/2 = 1/3 Die 1 wins
What is the probability that Die 1 rolls a 2 and Die 2 rolls a 5? Use the AND lemmato multiply probabilities
2/3 x 1/2 = 1/3 Die 2 wins
What is the probability that Die 1 rolls a 6 and Die 2 rolls a 1? Use the AND lemmato multiply probabilities
1/3 x 1/2 = 1/6 Die 1 wins
What is the probability that Die 1 rolls a 6 and Die 2 rolls a 5? Use the AND lemma to multiply probabilities
1/3 x 1/2 = 1/6 Die 1 wins
What is the probability that Die 1 beats Die 2? Use the OR lemma to add probabilities
1/3 + 1/6 + 1/6 = 2/3

These lemmas can be applied to die with any numbers as well as other probabilities

Lorenmartell 03:13, 7 December 2010 (UTC)