I was watching Veritasium's Would You Take This Bet? video. In a part of the video Derek asks people whether they would accept the bet in the case of each true guess for flipping the coin the person would win $10$ dollars and for each true guessing this $10 $ dollar will increase twofold as $10+20+40$ ... etc. But for each false guess the person betting will lose $10$ dollars. So in video he tells that probability of losing money for $100 $ times of guessing is $1/2300$ . I tried to find the this probability by myself. I mean it is obvious the probability of losing money in these circumstances but I couldn't find the same conclusion as Derek. So I tried to find the minimum number of trues guesses that would make the person at lose in final situation. For $10$ of his guesses are true $10\times ((1-(2^7))/(1-2))=1270\rightarrow$ Gaining money $10\times 93=930 \rightarrow$ Losing Money. For $9 $ of his guesses are true $10\times ((1-(2^6))/(1-2))=630 \rightarrow$ Gaining money $10\times 94=940 \rightarrow$ Losing Money. In order to lose money, the person has to be false at least $ 6$ of his guesses. pbinom(6,size = 100, prob = 0.5) = 1.00298e-21 This the result that I found. Where did I make a mistake?

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