# 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)