Harry, anything that helps a perosn understand probability in gambling I think is of some use and a worthwhile use of brain power.
The question on Double Up EV is more a question of semantics than probability. It's all in how you define the particular EV in which you are interested. If you are looking at the EV of all your money in play for a given 100 hand sample, adding Double Up will increase the overall percentage EV of the money you have in play for that session. Playing 100 hands at 99.73% return and 46 hands at 100% return will increase your overall average percentage return but since you are putting more money in play, you will win or lose the same amount.
Let's say you play 100 hands of NSUD ( 99.73%), and get the following hand distribution ( quick auto play on Winpoker), 1 SF, 9 quads, 4 full house, 1 flush, 4 Straight, 27 trips and 54 whiffs. You happen to break even on this particular short session.
Regardless of whether you double up or not, your results from the NSUD portion are the same. Unless you are one of those people who believes that when you hit the keys actually changes your expected results. That notion aside, the NSUD portion of the play only determines how often you have an opportunity to double up. The NSUD EV of the game is determined by the pay table. The Double Up EV of the game is determined by its 'paytable'. Regardless of whether you double up 1 time, 10 times or 46 times, the EV of the Double Up Feature is 100%. It is just a question of how often you double up.
In the above example, you have 500 coins wagered at an expected return on 99.73% so your expected coin out is 498.65 coins. You now also bet 500 coins on double up and your expected coin out from that bet is 500 coins.
For the 1000 coins wagered ( Double Up plus NSUD), your expected return is 998.65 coins or 99.865%. Seems like double up is a good idea. Unfortunately, you are risking twice as much money so your expected loss is the same 1.35 coins. All you have done is increase the amount in play and thus the variance.
I think it is a pretty interesting exercise ( my wife thinks I'm weird).
In summary, adding a 100% return play to a given, less than 100% play will increase the percentage return but not the actual coin return, due to increased wager size. It's the same argument as the odds on the pass line in dice.
Now , if the discussion is whether to play 200 hands of NSUD or 100 hands of NSUD with double up ( same expected coin in for both activities), that is a different discussion
[Non-text portions of this message have been removed]
