1. Dice Games & Probabilities
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Dice Games & Probabilities

2. 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 Class 16

3. 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 Class 16

4. Table of the 36 Possible Outcomes of Throwing a Pair of Dice
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Throwing a Pair of Dice
Table of the 36 Possible Outcomes of Throwing a Pair of Dice
Total Dots Combinations # Ways ,
, 3+1,
, 4+1, 2+3,
, 5+1, 2+4, 4+2,
7 1+6, 6+1, 2+5, 5+2, 3+4,
, 6+2, 3+5, 5+3,
, 6+3, 4+5,
, 6+4, ,
Total # Ways = 36 l
Thermo & Stat Mech - Spring Class 16

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) =

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

7. Thermo & Stat Mech - Spring 2006 Class 16
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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 Class 16

8. Example with Some Complications
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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 Class 16