# The probability of Breakeven On a Coin Toss Game

I was walking the other day around my work office in NYC and thought of this interesting scenario in a game of coin flips.

You have $500 in your pocket. This is your entire life savings. You play a game of coin flips with a two sided, fair coin, 5 times. Your winnings and your losses is the same as the capital you risk. So if you risk$$100, you can either win an addition$100 or lose the entire $100 you invested. You argue that since the odds are 1:1, it doesn't matter what you pick, so you choose tails for each of the 5 coin flips betting a$100 each, successively as the coins are flipped.

You play the game and the outcomes are all heads, HHHHH. You've lost your entire $500 of life savings. You have two options. 1) Walk away empty handed knowing your unlucky. Or 2) you go to a loan shark down the alley to loan you$100 to play the game one.. more..time. Who knows maybe you'll break your unlucky streak the 6th time around?

Question is, should you take a loan and play the game 1 more time to even things out knowing the 6th coin flip is still 50-50 and completely independent from the last 5?

• Ha! Let's forget about the consequences if we don't pay the loan shark for the moment and go the heart of the question, should you keep betting? The probability of a H-T on each flip 50-50. So I can understand that picking Tails on each coin flip shouldn't make a difference. OR...... you can look at from cumalitive perspective. What is the probability I get TTTTT on 5 coin flips, Ans(1/32). What is the probability of getting p(HHHTT), answer 10/32. – jessica Dec 18 '14 at 22:47

Knowing that the 6th coin flip is still 50-50 and completely independent from the last 5 should make the entire intro of your question void. You should ask - is it worth borrowing \$100 in order to have a 50-50 chance of winning an additional \$100. Everything else is just just noise.

To answer this just think about the simple goal of maximising the probability of future prosperity. So if you owe another lender \$100 and they will kill you in an hour if you don't repay them - then take the risk and you have a 50-50 chance of living (yay!). However, if you are not in harsh debt and the loan shark will kill you if you don't pay them back tomorrow then you are taking a 50-50 chance of getting murdered (bad idea!). I know this was not a very 'mathsy' answer, but neither was the question :P • Oh by the way I assumed zero interest! Which is probably unreasonable as @paw88789 suggested – j__ Dec 18 '14 at 22:29 • Ha! Ya, lets assume no harm can be done to you and the interest is a fair one. Let's just ignore that in this case. – jessica Dec 18 '14 at 22:41 • @jessica - If you are interested in this kind of thing (how to make decisions) there is a field of study called 'game theory' which has considered for a long time and looks at problems that require optimal decision making. Any basic book in the area should offer tons of these types of questions and offer frameworks for answering them. You might find it a fun read over the holidays! Anyway if you think I answered the question appropriately could you accept it please - thank you :) – j__ Dec 18 '14 at 22:46 • This answer correctly points out there may be a difference between how you value gaining or losing the money, and @jackF, yes, game theory is one place to look for more details on this. So is a good microeconomics book. – Trurl Dec 18 '14 at 23:40 My strategy. First notice that the expected return in this game in the long run is zero dollars. You win what you risk, and you would expect to win and lose with roughly equal frequency. If your bets are constant, you should come pretty close to break even in the long run. But that might mean going way into to debt for a while and then slowly winning your way back up to even. However, there is a better strategy, assuming there is no limit to what you can bet. Always bet double your losses. Go take out an enormous loan. Now, you're down \$500 plus the loan fees and short term interest -- let's say all that amounts to \$2000. So bet \$4000. If you win, return the loan in it's entirety and walk away. If you lose (you'd be down \$6k), so bet \$12,000 on the next flip. Again: win and walk away or lose and re-double. If at any point you win and walk away you'll walk away up on earnings. Pay off the loan and never play the game again.

This strategy is essentially the reason most games that involve gambling also involve limits to the bets. In a no-limit betting situation you could follow this strategy to pretty much insure yourself against losses. The caveat is that you might need an incredibly large amount of money to continue the doubling until you win. It wouldn't take long for millions of dollars to be at stake, and if at any point you are unable to bet more than your losses the strategy fails and you have dug yourself into extreme debt.

## But, of course you shouldn't actually play the game just one more time.

Obviously. no. Don't play. Walk away. Whatever you have to pay for the loan is additional losses, which makes the expected value of playing the game one more time negative, even if slightly so. Let's say you put One kajillion dollars on the table. You have a 50% chance of losing the kajillion dollars and a 50% chance of winning the kajillion dollars. So, expected value: \$0. But now take into account the loan fees and we are in negative territory. If you win the bet, you get the kajillion dollars. You are a kajillionaire (minus your original losses and the fees for the loan which you can quickly pay off). If you lose, you are a negative kajillionaire minus the fees for the loan and your original losses. Let$k = \text{1 kajillion}, l = \text{losses so far}, f = \text{loan fees + interest}\$. The average of those two outcomes is

\begin{align*} \dfrac{(k - l - f) + (-k - l - f)}{2} &= \dfrac{-2(l + f)}{2} = -(l + f) \end{align*}

Meaning that playing again you should view as having roughly the same cost as simply walking into the loan office and paying the fees on a loan you don't take out. So, you don't play again.

• When you say "an incredibly large amount of money to continue the doubling until you win", its worth noting that any finite amount of money leaves you with a likelihood of going bankrupt before a win - as there are a finite number of dollars in circulation this becomes a losing strategy unfortunately :) – j__ Dec 19 '14 at 0:06
• @jackf is credit also hamstrung with physical limitations? Maybe back the bets with IOU's? If you lose all the money in the world, at least you're committed to the strategy. – Nick H Dec 19 '14 at 0:19

I would say you should not. Considering that the loan shark surely charges an unreasonably high interest rate, your expected value for this sixth flip is negative. Don't do it. And if you lose and can't repay the shark, you may end up with hospital bills too from the broken legs.

Just go back to working hard and try to make some more money.