Variance using Double Up on Triple Bonus Plus

Hi,

Triple Bonus Plus has an EV of ~.998 (yum) and a variance of ~44.28 (yuk). Let's say I find that machine where the double up feature is enabled, and I can take my wins and use the double up feature 4 times. I think I can keep the payback the same and reduce the variance. Can you guys check my math / thinking?

I have to go look at the win / loss rate on this game using a given strategy, but I'll assume it is 50%. (does anyone know where to find the hit rate for video poker?)

If I have the hit rate right, half the time (when I lose), I am playing with a variance of 44.28.

When I win, I use the double up feature up to 4 times (as long as I keep winning).
I think the double up feature should have about the same variance as a coin flip ~.25? (Does anyone know where the variance is for the double up feature in video poker?)

So...
12.5% of the time, I have variance of 9.06 (.25 + .25 + .25 + .25 + 44.28) / 5
12.5% of the time, I have variance of 11.26 (.25 + .25 + .25 + 44.28) / 4
12.5% of the time, I have variance of 14.93 (.25 + .25 + 44.28) / 3
12.5% of the time, I have variance of 22.27 (.25 + 44.28) / 2

So I I take the weighted average, I get a total game variance of 29.33 using the double up 4 time on a winning hand strategy...
Also, I believe since there is no house edge on the double up, that the total return to player would be unchanged.

Am I thinking about variance and this game correctly?

If this belongs in the math section, I apologize.

You're thinking about it in the exact opposite way. Let's say the game is played at the 25c denom and you hit a royal. You double 4 times so it goes from 1k to 2k, to 4K, 8k, then 16k. You'll only get the 16k every 1 in 16. Does that seem like it increases or decreases the variance?

Thanks for the response. I was hoping you would reply!

Intuitively it seems like variance should increase as you stated. I had a feeling I was putting the math together wrong. What is the right way to combine a high variance event (like a hand of triple bonus plus) with a low variance event (like a coin flip or 4) to get total variance?

I'm going to Google calculating probably & variance of rolling a die and flipping a coin. I know they covered that in college. Sadly that was a long time ago!

Generate the strategy on WOO. The strategy isn't going to be changing, but you're going to be looking at the COMBINATIONS column. I think this is the game you referenced, but if not, the same theory applies, just switch the game and the respective numbers.

https://wizardofodds.com/games/video-poker/strategy/a-1-b-118-c-1-d-0-d-1-d-1-d-3-d-4-d-5-d-9-d-50-d-120-d-240-d-100-d-800/

Royal Flush -- pays 800 -- 513,460,056 combinations

If you double up 4 times, then you're effectively only hitting a royal flush 1/16 of the time above. 2^4 = 16. Royal would pay 12,800 for 1 (instead of 800 for 1).

513,460,056/16 = 32,091,253.5

The remainder of the time (513,460,056 - 32,091,253.5 = 481,368,802.5) the payout is going to be 0. Take that number (~481million) and add it to the "Nothing" combinations. That'll be 11,021,496,418,956+481,368,802.5 = 1.1021978e+13. So now, you should have the following:

Royal Flush -- pays 12,800 (for 1) -- 32,091,253.5 combinations
...
Nothing -- pays 0 -- 1.1021978e+13 combinations

Then do the same thing for all the rest of the hands. This would actually be fairly easy to do in Excel with a few formulas setup. I forget how to calculate the variance exactly on VP (I always do it wrong). But if you do the above, then you'll have the correct combinations and pays.