Mathematicians needed to solve this question

[Heisenberg]

I need a better person at maths than me to explain the solution to the following issue.

Say you're an AP with an edge of about 0.5% to 1% and you are able to spend as much time in a casino as you wish (important point).

Say you're betting with a bankroll of $10,000 and you're playing at a $5 table using a spread of 1-10.

Say you go in to a casino and on the first shoe you find yourself up $500 by the end of the shoe. Can you prove mathematically that it's either (a) better to stop and move to another table, (b) continue or (c) it makes no difference...

My intuition says it would be best to move to another table as you've hit the table at a lucky moment in time which, if you continue is likely to re-balance and you'll gradually lose some of your winnings back given enough time or tread water until you get down to your 0.5-1.0% edge, therefore wasting your time....

If my intuition is incorrect, please provide a proof.

Any help provided would be greatly appreciated...thanks

[Norm]

If you break a window in a jet at cruising altitude, air will rush out of the plane until the air pressure in the plane balances with the air pressure outside the plane. BUT, there is no such physical pressure in games involving independent trials. The cards don’t know that you won or lost in the past and will not make an effort to “re-balance”.

[Banker]

Assuming your edge still the same, there is no reason to change tables. The edge is for your "long term" win and not specific to any given table or specific sets of cards, as Norm explained.

[Heisenberg]

The cards don’t know that you won or lost in the past and will not make an effort to “re-balance”.

I wasn't intending to imply that. But over time there will be a tendency to approach your theoretic edge resulting in a '"re-balancing" which is what I meant to imply.

So you'd say (c) is the answer ?

[DSchles]

I wasn't intending to imply that. But over time there will be a tendency to approach your theoretic edge resulting in a '"re-balancing" which is what I meant to imply.

So you'd say (c) is the answer ?

If you toss a coin 100 times, you're "supposed" to expect that 50 heads will be more likely than any other number. So, if you've tossed 90 times and already achieved those 50 heads, do you suppose the next 10 flips need to be tails to "rebalance" things? Think about how silly such a notion would be.

Don

[Banker]

The theoretic edge would be approached by increasing the number of trials (long run) but it is not specific for a table or even for a given person.

[Heisenberg]

If you toss a coin 100 times, you're "supposed" to expect that 50 heads will be more likely than any other number. So, if you've tossed 90 times and already achieved those 50 heads, do you suppose the next 10 flips need to be tails to "rebalance" things? Think about how silly such a notion would be.

Don

That's also not what I was implying as you've set an arbitrary limit of 100 coin flips...

[Heisenberg]

The theoretic edge would be approached by increasing the number of trials (long run) but it is not specific for a table or even for a given person.

You're right of course and this is what my head tells me. However, my intuition tells me otherwise and historically my intuition is often correct

[RS]

I wasn't intending to imply that. But over time there will be a tendency to approach your theoretic edge resulting in a '"re-balancing" which is what I meant to imply.

So you'd say (c) is the answer ?

RE bolded -- the way you're likely interpreting that is false.


Over time, you expect to get closer to your expected PERCENTAGE win, not ACTUAL (sort of). Break it down into a simpler problem, a coin flip which is 50/50.

For starters, if you flip a coin 1,000 times you expect 500 heads and 500 tails.


You flip it 100 times and get 56 heads and 44 tails. Let's say the heads are wins and tails are losses, meaning so far you've won at a rate of 12% of your action (56 wins - 44 losses = +12 net, divided by 100 = +12%). Of course, the expected value is 0% (50 wins - 50 losses = 0 net, divided by 100 = 0%).

If you flip the coin 900 more times, of those 900 flips, you'd expect 450 heads and 450 tails. So, you expect your ending result to be 450+56 heads (506) and 450+44 tails (494). At this point, you're still up 12 units, for a net win % of 12/1,000 = 1.2%.


As you can hopefully see, you expect your results to tend towards the expectation as a percentage (from 12% to 1.2%), but the actual amount won't "dip downwards" (as if 12 headed toward 0).



The past doesn't matter on your future results, assuming nothing changes about the game itself (worsening conditions, etc.). It doesn't matter if I go into a session with a $10K bankroll, $100K bankroll, or a $1M bankroll....if I'd be playing the same game in all those cases, our expectation is going to be the same.

[Banker]

You're right of course and this is what my head tells me. However, my intuition tells me otherwise and historically my intuition is often correct

You are not alone in this. In fact, most people would feel the same way. That is how casinos make so much money by putting up past roulette numbers. Most people will think if it hits black five times in a row, it has a greater chance of hitting red but it is not correct.

[Heisenberg]

The past doesn't matter on your future results

That is the key of course, though not always true in blackjack because of how the game is played (hence shuffle tracking)...

[Heisenberg]

You are not alone in this. In fact, most people would feel the same way. That is how casinos make so much money by putting up past roulette numbers. Most people will think if it hits black five times in a row, it has a greater chance of hitting red but it is not correct.

Whilst I agree with that, BJ is a very different game to roulette and can't really be compared...

[RS]

That is the key of course, though not always true in blackjack because of how the game is played (hence shuffle tracking)...

Everything has an exception. You could compare blackjack to roulette because if you saw a certain number (or section/sector) show up more likely than expected, you might have stumbled upon a wheel with bias.