# What Is the Probability of Losing Money?

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?

Instead it is a fair coin with favourable bets (win $$+20$$, lose $$-10$$). So if you have $$100$$ bets, you will lose money with if your side comes up $$33$$ or fewer times out of $$100$$ but win overall with your side coming up $$34$$ or more times, since $$20 \times 33 -10 \times 67 =-10 <0$$ while $$20 \times 34 -10 \times 66 =+20 > 0$$.
That makes the probability of losing overall $$\sum \limits_{k=1}^{33} {100 \choose k} 2^{-100}$$ which you can find in R with pbinom(33,100,1/2) giving about $$0.00043686$$ or about $$\frac{1}{2289}$$.
• If you were to say that with your side coming up $6$ times you would get $10+20+40+80+160+320 -10\times 94 = -310<0$ and coming up $7$ times $10+20+40+80+160+320 +640-10\times 93 = +340>0$ then yes, pbinom(6,100,1/2) giving about $10^{-21}$ would be the equivalent calculation. Mar 12, 2023 at 12:55