Dice Games Probabilities Dice Games One die has
Dice Games & Probabilities
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. • Its equally probable that any number • from 1 to 6 will come up. l Thermo & Stat Mech - Spring 2006 Class 16
• 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
• 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 outcomes NP! l Thermo & Stat Mech - Spring 2006 Class 16
Throwing a Pair of Dice Table of the 36 Possible Outcomes of Throwing a Pair of Dice Total Dots 2 3 4 5 6 7 8 9 10 11 12 Combinations 1+1 1+2, 2+1 1+3, 3+1, 2+2 1+4, 4+1, 2+3, 3+2 1+5, 5+1, 2+4, 4+2, 3+3 1+6, 6+1, 2+5, 5+2, 3+4, 4+3 2+6, 6+2, 3+5, 5+3, 4+4 3+6, 6+3, 4+5, 5+4 4+6, 6+4, 5+5 5+6, 6+5 6+6 Thermo &# Stat. Ways Mech - Spring 2006 = 36 l. Total Class 16 # Ways 1 2 3 4 5 6 5 4 3 2 1
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 l Thermo & Stat Mech - Spring 2006 Class 16
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
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
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 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? Solution l Thermo & Stat Mech - Spring 2006 Class 16 11
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 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 Problem 3 • 2 dice are thrown, what is the probability of getting only one 6? 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? 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 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 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 Thermo & Stat Mech - Spring 2006 Class 16 15
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