Probability Question? Could be easy ( apologies) but may not be?

Suppose you have an initial balance of £50 and play £10 stakes on a coin- heads you win ( and gain £10) / tails you lose. What is the probability of doubling your initial balance assuming you play as long as you need to? Or indeed the probability of losing the balance?
How does the stake size affect the probabilities? What if you played at £5 will it make any difference? And what if i went for a target of £150


Also generalising, balance of B
Stake size - s
target - T ( 2xB or 3xB etc)
p= probability of a win on one trial?

Anyone got any ideas? Its either really simple or really complicated??
Many Thanks

With a coin you are just as likely to win as you are to go bankrupt.

With craps pass line bets (no odds) and the £10 bet you have a 46.47% of doubling your money before you go bankrupt.
With the £5 bet you have 42.98% of doubling before going bankrupt. A pass line bet has a HA of 7/495 = 1.41%.

With £10 coins and trying to make it to £150 you have 28.75% of tripling before you go bankrupt.


The formula is in the Wizard's math problems. It is pretty simple.

In general you always increase your expectation by reducing your number of bets. But you greatly increase your variance. If you reduce it down to betting your entire bankroll in one bet you will have the best expectation, but you can easily lose everything at once.

Thanks for the quick reply. Can you give me an idea where i can find the formula- I have had a look- Sorry not being lazy just can't find it.

Modify this solution to your general problem .

Quote: rodders

Suppose you have an initial balance of £50 and play £10 stakes on a coin- heads you win ( and gain £10) / tails you lose. What is the probability of doubling your initial balance assuming you play as long as you need to? Or indeed the probability of losing the balance?
How does the stake size affect the probabilities? What if you played at £5 will it make any difference? And what if i went for a target of £150


Also generalising, balance of B
Stake size - s
target - T ( 2xB or 3xB etc)
p= probability of a win on one trial?

Anyone got any ideas? Its either really simple or really complicated??
Many Thanks

There's a general rule for an even-money bet with equal probability to win or lose.
The probability of reaching some win goal (WG) before reaching a loss limit (LL) is:

p(WG) = LL / ( WG + LL)

So, if you want to win $50 and are willing to lose $100:

p(WG) = 100 / (50 + 100) = .333
etc.

I don't find it very useful, because there is no time factor; it assumes you play until you either bust or reach the goal, but this could take a very, very long time.
Cheers,
Alan Shank