Dice Games & Probabilities 3/21/06 Dice Games & Probabilities

Dice Games NOT Binomial Distributions! Problem: Solution: 3/21/06 Dice Games One die has 6 faces. So, the probabilities associated with a dice game are NOT Binomial Distributions! For 1 die, the probability of any face coming up is p = 1/6. So, its 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 ≡ Np. Then, we must count the number of those that give the desired outcome ≡ no. l Thermo & Stat Mech - Spring 2006 Class 16

We first must list all possible outcomes NP! 3/21/06 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 ≡ Np. Need to also count the number of those that give the desired outcome ≡ no. The probability of the desired outcome = number that gives the desired outcome divided by the total number of outcomes. P(no) = (no)/(Np) So, p = 1/6 for one die. To do this for a pair of dice, We first must list all possible outcomes NP! l Thermo & Stat Mech - Spring 2006 Class 16

Table of the 36 Possible Outcomes of Throwing a Pair of Dice 3/21/06 Throwing a Pair of Dice Table of the 36 Possible Outcomes of Throwing a Pair of Dice Total Dots Combinations # Ways 2 1+1 1 3 1+2, 2+1 2 4 1+3, 3+1, 2+2 3 5 1+4, 4+1, 2+3, 3+2 4 6 1+5, 5+1, 2+4, 4+2, 3+3 5 7 1+6, 6+1, 2+5, 5+2, 3+4, 4+3 6 8 2+6, 6+2, 3+5, 5+3, 4+4 5 9 3+6, 6+3, 4+5, 5+4 4 10 4+6, 6+4, 5+5 3 11 5+6, 6+5 2 12 6+6 1 Total # Ways = 36 l Thermo & Stat Mech - Spring 2006 Class 16

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

Probabilities for Throwing Two Dice 3/21/06 Probabilities for Throwing Two Dice l Thermo & Stat Mech - Spring 2006 Class 16

Thermo & Stat Mech - Spring 2006 Class 16 3/21/06 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 l Problem 2 Two normal dice are thrown. What is the probability of two 6’s coming up? l Thermo & Stat Mech - Spring 2006 Class 16

Example with Some Complications 3/21/06 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 1st die & not the 2nd & the probability of the 6 on the 2nd die & not the 1st are both equal to So, the probability of getting only one 6 is: l Thermo & Stat Mech - Spring 2006 Class 16