I believe this is a fairly simple question but am unsure as to what the correct approach is.
Suppose you flip a fair coin and bet ${$}x$. If heads comes up, you win ${$}.9x$ ; if tails comes up, you lose ${$}x$. You have ${$}10,000$ and you need to win at least ${$}5,000$. What is the likelihood of winning at least ${$}5,000$ with a single bet? What is the likelihood of winning at least ${$}1,000$ by playing 10,000 times and o only betting ${$}1$ each time? What is the likelihood of not losing money?
My confusion arises as to what probability I’m supposed to use. The ‘fair’ probably since it’s a fair coin, or the dealers implied probability based on the winnnings and loses?
For the first part of the question, it's a single bet, meaning that you bet all $10,000$ dollars, and you either win $0.9 * 10000 = 9000$ dollars or you lose $10,000$ dollars. There's 2 outcomes with equal probability and only in one of them you make at least $5,000$ dollars.
For the second and third part of the question, you're doing $10,000$ flips with a bet of one dollar one each. Each trial is still a flip on a fair coin, so there's a 50% chance you make 90 cents and a 50% chance you lose the dollar. In this way, you can think of it as a binomial distribution with 10,000 trials and P(Success) of 0.5. You can also calculate n, the number of trials you need to win to make at least 1,000 dollars and move on from there.