Payout identity, expectancy, and path-dependent ruin

A stake-excluded-reading of the advertised payout

A reader treated a 70 percent advertised payout on a 10-unit stake as a 7-unit winning credit and omitted return of the original stake. That treatment is a stake-excluded-reading of the payout-identity.

Under that stake-excluded-reading, 500 wins of 7 units and 500 losses of 10 units were tallied as 3500 against minus 5000, leaving minus 1500.

The same stake-excluded arithmetic was said to require 590 wins in 1000 trials, credited as 4130, to offset 410 losses of 10 units, or minus 4100, leaving a remainder of 30. That required win count is the breakeven-hit-rate under the stake-excluded-reading.

The cash a winning settlement actually credits

A reply stated that a winning settlement returns the original 10-unit stake plus the 70 percent increment, for 17 units credited on each win. That statement names the payout-identity, including return of the original stake.

Under the stake-included identity, 500 winning settlements were tallied as 17 times 500, or 8500, rather than 7 times 500.

The published figure was described as a net-payout-schedule and not as a profit-and-loss table. A net-payout-schedule is a table of cumulative settlement cash. It is not the same object as a marked-to-market profit-and-loss statement.

Mixed-outcome increment after the identity is named

Expected-value is the average increment implied by the payout-identity, the stake size, and a stated mix of winning and losing trials. Terminal-expectancy is the net increment implied by a completed count of wins and losses, regardless of the order in which those outcomes arrived.

Editorial reading applies expected-value only after payout-identity is named. The stake-excluded-reading produced a minus 1500 remainder on 500 wins and 500 losses. The stake-included identity credits 8500 on those 500 wins. Those two tallies are not interchangeable filters.

Hostile order and path-dependent-ruin

From a 1000-unit starting balance, 500 wins followed by 500 losses were described as finishing 3500 above the opening deposit. Editorial reading treats that completed count as terminal-expectancy for the mix, not as a statement that every order survives.

From a 5000-unit starting balance, 500 losing trials arriving first were described as exhausting the account before later wins could be collected. That exhaustion is path-dependent-ruin.

Risk-of-ruin is the chance that a bounded account hits zero, or cannot continue, before a planned sequence of trials is finished.

A random mix of the same 500 wins and 500 losses was described as still able to finish near a 3500 increment after 1000 trials if the account remained intact. Monte-carlo-simulation is repeated random reorderings of the same win and loss counts to see which paths survive and which exhaust the account.

Editorial reading treats that third check as a separate filter. Terminal-expectancy does not answer whether a finite account can walk the same outcomes in a hostile order without hitting ruin.