1. Dice Games & Probabilities

2. Thermo & Stat Mech - Spring 2006 Class 16 Dice Games l One die has 6 faces. So, the probabilities associated with a dice game are NOT Binomial Distributions! When throwing 1 die, the probability of any face coming up is p = 1/6. So, it is equally probable that any number from 1 to 6 will come up. Problem: When throwing 2 dice, what is the probability that the total will come up 2, 3, 4, etc up to 12? Solution: To calculate the probability of a particular outcome, we must first count the number of possible outcomes ≡ N p. Then, we must count the number of those that give the desired outcome ≡ n o.

3. Thermo & Stat Mech - Spring 2006 Class 16 l When throwing 2 dice, what is the probability that the total will come up 2, 3, 4, etc up to 12? Solution: Need to count the number of possible outcomes ≡ N p. Need to also count the number of those that give the desired outcome ≡ n o. The probability of the desired outcome is equal to the number that gives the desired outcome divided by the total number of outcomes. P(n o ) = n o /N p So, p = 1/6 for one die. To do this for a pair of dice, We first must list all possible outcomes N P !

4. Thermo & Stat Mech - Spring 2006 Class 16 Throwing a Pair of Dice l Table of the 36 Possible Outcomes of Throwing a Pair of Dice Total Dots Combinations # Ways 2 1+11 3 1+2, 2+12 4 1+3, 3+1, 2+23 5 1+4, 4+1, 2+3, 3+24 6 1+5, 5+1, 2+4, 4+2, 3+35 71+6, 6+1, 2+5, 5+2, 3+4, 4+36 8 2+6, 6+2, 3+5, 5+3, 4+45 9 3+6, 6+3, 4+5, 5+44 10 4+6, 6+4, 5+53 115+6, 6+52 12 6+61 Total # Ways = 36

5. The Probability Model for Two Fair Dice Example of a Random Phenomenon: Roll pair of fair dice. The Sample Space is illustrated in the figure: The probabilities of each individual of the 36 outcomes are found by inspection. Each clearly occurs with a probability of p = (1/36) = 0.0278

6. Thermo & Stat Mech - Spring 2006 Class 16 6 Probabilities for Throwing Two Dice l

7. Thermo & Stat Mech - Spring 2006 Class 16 7 Examples Problem 1 Two faces of a die are painted red. When the die is thrown, what is the probability of a red face coming up? Solution P Problem 2 Two normal dice are thrown. What is the probability of two 6’s coming up? Solution l

8. Thermo & Stat Mech - Spring 2006 Class 16 8 Example with Some Complications p = probability of success (p = 1/6 for 1 die). q = probability of failure (q = 5/6 for 1 die). Of course p + q = 1, or q = 1 – p Problem 3 2 dice are thrown, what is the probability of getting only one 6? Solution The probability of the 6 on the 1 st die & not the 2 nd & the probability of the 6 on the 2 nd die & not the 1 st are both equal to So, the probability of getting only one 6 is: l