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# Dice Games & Probabilities

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1. Dice Games & Probabilities

2. One die has 6 faces. So, the • probabilities associated with a dice • game are • NOTBinomial Distributions! • For 1 die, the probability of any face • coming up is p = 1/6. • Its equally probable that any number • from 1 to 6will come up. Dice Games l Thermo & Stat Mech - Spring 2006 Class 16

3. For 1 die, the probability of any face coming up is • p = 1/6. • 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

4. When throwing 2 dice, what is the probability • that the total will come up 2, 3, 4, etc up to 12? • Solution: • 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 outcomesNP! l Thermo & Stat Mech - Spring 2006 Class 16

5. Throwing a Pair of Dice Table of the 36 Possible Outcomes of Throwing a Pair of Dice Total DotsCombinations # 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

6. Example of aRandom Phenomenon:Roll pair of fair dice. The Sample Spaceis illustrated in the figure: Probability Model for Two Fair Dice 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

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

8. 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? P l l Thermo & Stat Mech - Spring 2006 Class 16

9. 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 l Thermo & Stat Mech - Spring 2006 Class 16

10. 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

11. 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? Solution l Thermo & Stat Mech - Spring 2006 Class 16

12. 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 l Thermo & Stat Mech - Spring 2006 Class 16

13. 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? l Thermo & Stat Mech - Spring 2006 Class 16

14. 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 the6 on the 2nd die & not the 1st are both equal to l Thermo & Stat Mech - Spring 2006 Class 16

15. 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 the6 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