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Probability

This guide explores the fundamentals of probability using dice. It explains the chances of rolling various outcomes with one or more dice, such as rolling a six or multiple sixes, using basic probability formulas. We delve into calculations like the probability of rolling exactly two sixes with three dice and the use of binomial coefficients to determine probabilities involving multiple trials. The explanation simplifies complex concepts, making it easier for learners to grasp how probabilities work in the context of dice.

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Probability

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  1. Probability Using dice

  2. One Die Chance of rolling a 6 with single die p = 1/6 Chance of not rolling a 6 p’ = 1-p

  3. Two Die Chance of rolling double 6’s with two die p = (1/6)(1/6) = 1/36 Chance of not rolling a double 6’s p’ = 1-p

  4. Three Die, Three Sixes Chance of rolling all 6’s with three die p = (1/6)(1/6)(1/6) = 1/216 Chance of not rolling all 6’s p’ = 1-p

  5. n Die, n Sixes Chance of rolling all 6’s with n die p = (1/6)n Chance of not rolling all 6’s p’ = 1-p

  6. Three Die, Two Sixes Chance of rolling exactly two 6’s with three die p =(1/6)(1/6)(5/6)? No! dice aren’t ordered This is probability of rolling double 6’s with two die followed by not a six on the third die. We don’t care whether first, second, or third die is not a 6

  7. Three Die, Two Sixes Chance of rolling exactly two 6’s with three die We don’t care whether first, second, or third die is not a 6 so: p =(5/6)(1/6)(1/6)+(1/6)(5/6)(1/6)+(1/6)(1/6)(5/6) Or p = 3 (1/6)(1/6)(5/6) = 15/216

  8. Three Die, Two or Three Sixes At least two 6’s with three die Don’t care about third die! 3 * (1/6)(1/6)(6/6) ? No! Double counts trip 6’s

  9. Three Die, Two or Three Sixes At least two 6’s with three die (1/6)(1/6)(1)+2(5/6)(1/6)(1/6) = 1/36+10/216 = 16/216 = 2/27 Or easier: (Two 6’s or Three 6’s) 3(5/6)(1/6)(1/6)+(1/6)(1/6)(1/6) = 15/216 + 1/216 = 16/216 = 2/27

  10. What About n Die, x Sixes? Let p = 1/6 Let q = 1- p Let B be the number of ways we can choose exactly x sixes out of a population of n die = B (px q(n-x))

  11. What About n Die, >= x Sixes? Let p = 1/6 Let q = 1- p Let Bi be the number of ways we can choose exactly i sixes out of a population of n die Sum for all i in x..n { (Bi (pi q(n-i)) }

  12. Binomial Coefficient Given population n Given x samples (nx ) = BiCoef(n,x) = n! / x!(n-x)! http://mathworld.wolfram.com/BinomialCoefficient.html

  13. Chose 2 from 4 Chose 2 from { A, B, C, D } {A, B}, {A, C }, { A, D }, {B, C }, {B, D }, {C, D} BiCoef(4,2) = 6 4! 2!(4-2)!

  14. Chose 3 from 5 Chose 3 from { A, B, C, D, E } ABC,ABD,ABE,ACD,ACE,ADE,BCD,BCE,BDE,CDE BiCoef(5,3) = 10 5! 3!(5-3)!

  15. Chose 3 from 6 Chose 3 from { A, B, C, D, E, F } BiCoef(6,3) = 20 6! 3!(6-3)!

  16. Chose 4 from 8 Chose 4 from { A, B, C, D, E, F, G, H } BiCoef(8,4) = 70 8! 4!(8-4)!

  17. Example 35 users Each user active 10% of the time Find probability that 10 users are active BiCo(35,10) (.1)10(.9)25

  18. Example 35 users Each user active 10% of the time Find probability at least 10 are active BiCo(35,10) (.1)10(.9)25 +BiCo(35,11) (.1)11(.9)24 +… Or 1 - probability less than 10 are active Probability at least 10 are active + probability less than 10 are active = 100% BiCo(35,9) (.1)9(.9)26 +BiCo(35,8) (.1)8(.9)27 +…

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