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lecture_02
pg.16 # 3, 7, 8, 14 pg.18 # 8, 9, 12, 13
Read 1.1 - 1.5
[{\left( {1 + x} \right)^h} = \sum\limits_{k = 0}^n {n \choose k} x^k ]
[ {n \choose k} = \frac{ n!}{ k!( n - k)!}]
[{\left( {1 + x} \right)^h} = (1 + x)(1 + 3x + 3x^2 + x^3)]
[=] [1] [3x] [3x^2] [x^3]
[ {n \choose k} + { n \choose k+1} = {n+1 \choose k+1}]
[ \frac{ n!}{ k!( n-k)!} + \frac{ n!}{ (k+1)!(n-k-1)!} = \frac{ n!}{k!(n-k-1)!} \left( \frac{ 1}{ n-k} + \frac{ 1}{ k+1} \right)]
[ \frac{ (n+1)!}{ (k+1)!(n-k)! } = {n+1 \choose k+1}]
2600 years ago: many facts
"theorems from axioms"
2300 years ago (Euclid).
Accurate facts about odds in gambling known around 17th centry
"theorems from axioms" about 1930 (kolmogovov)
Chevalrer de Mere c.1654
his bet: I can get a 6 on a six sided dice in four tosses of a die.
His calculation: [ 4 \times \frac{ 1}{ 6} = \frac{ 2}{ 3}]
[prob( Win) + prob( Loose) = 1 ]
[p + \frac{ 5}{ 6}^4 = 1]
[ p = 1 - \frac{ 5}{ 6}^4 = 1 - \cdots ]
An immortal plays a game in which his chance of winning is [ \frac{ 1}{ 3} ]. If he plays once per day what is the probability he eventually wins?
[ p( win) = \sum\limits_{k = 0}^\infty \frac{ 1}{ 3} {2 \choose 3}^k = 1 ]
[ (1 + x)^{-1} = 1 - x + x^2 - x^3 +x^4 - x^5 \cdots ]
[ (1 - x)^{-1} = \frac{ 1}{ 1 - x} = 1 + x + x^2 + x^3 + x^4 + \cdots ] Memorize