Gambler's ruin based on Harvard Fat Chance

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Gambler's Ruin

key questions

what if we don't know how many trials to do, then we can't use bernoulli or binomial trials

how to find out the probability of winning, only given p(win), a(initial capital), b(capital goal), a down to 0 = end game

Video links

original course p65-66

my note videos p61-62

chance of winning = 48%, chance of losing = 52%

either bet once with 1000 dollars with 48% chance winning, or bet one dollar at a time with 48% chance winning; which one is more riskier or which make you more likely to win?

x(a) = probability to win away b with initial money a before lose all a

how to construct x(a), we don't know how many games to try, we can't use bernoulli or binomial

but we can use conditional probability with total probability of sum to construct this x(a)

x(a) = P(win away b) = P(win away b & first bet won) + P(win away b & first bet lose)

• = P(first bet won) x P(win away b | first bet won) + P(first bet lose) x P(win away b | first bet lose)
• = $p \times x(a+1) + q \times x(a-1)$

to simply the formula above

practice