Dice Games & Probabilities

Thermo & Stat Mech - Spring 2006 Class 16

Dice Games

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•One die has 6 faces. So, the probabilities associated witha dice game are

NOT Binomial Distributions!•When throwing 1 die, the probability of any face comingup is p = 1/6. So, it is equally probable that any numberfrom 1 to 6 will come up.

Problem:•When throwing 2 dice, what is the probabilitythat the total will come up 2, 3, 4, etc up to 12?

Solution: •To calculate the probability of a particular outcome, wemust first count the number of possible outcomes ≡

Np.•Then, we must count the number of those that give

thedesired outcome ≡ no.

Thermo & Stat Mech - Spring 2006 Class 16l

•When throwing 2 dice, what is the probability that thetotal 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 thedesired outcome ≡ no. •The probability of the desired outcome is equal to thenumber 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!

Thermo & Stat Mech - Spring 2006 Class 16

Throwing a Pair of Dice

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Table of the 36 Possible Outcomes of Throwing a Pair of Dice Total Dots Combinations # Ways

2 1+1 13 1+2, 2+1 24 1+3, 3+1, 2+2 35 1+4, 4+1, 2+3, 3+2 46 1+5, 5+1, 2+4, 4+2, 3+3 57 1+6, 6+1, 2+5, 5+2, 3+4, 4+3 68 2+6, 6+2, 3+5, 5+3, 4+4 59 3+6, 6+3, 4+5, 5+4 410 4+6, 6+4, 5+5 311 5+6, 6+5 212 6+6 1

Total # Ways = 36

The Probability Model for Two Fair DiceExample 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

Thermo & Stat Mech - Spring 2006 Class 16

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Probabilities for Throwing Two Dice

2.8 5.6 8.3 11 14 17 14 11 8.3 5.6 2.8 %

361

362

363

364

365

366

365

364

363

362

361

Prob.

12 11 10 9 8 7 6 5 4 3 2 Total

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Thermo & Stat Mech - Spring 2006 Class 16

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ExamplesProblem 1

•Two faces of a die are painted red. When the die isthrown, what is the probability of a red face coming up?

Solution

P

Problem 2•Two normal dice are thrown. What is theprobability of two 6’s coming up? Solution

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31

61

61 p

361

61

61

)2( p

Thermo & Stat Mech - Spring 2006 Class 16

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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:

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365

65

61 pq

185

3610

2)1( pqp