|
|
|
January 5th, 2009
 | 01:38 pm - The answer:
Here's my answer to the optimum strategy question:
Spinner 1 should:
Spin on 0.70 or less and stay on 0.75 and above
Spinner 2 should:
Spin if trailing player one or if first spin is 0.60 or less
Spin if tied with player one and spin is 0.70 or less
Spinner 3 should:
Spin if behind player 1 or 2
Spin if tied with both players and first spin is 0.65 or less
Spin if tied with one player and first spin is less than 0.50 (odds are equal on 0.50 and player can spin or stay with the same expected results)
I am still not ready for the drawing and have a few more chances to give out depending on discussion here. Everyone who participated will be in the drawing. What I will do is post assigned wheel spaces to all of you and a show date. The 1st spin of the show that airs on WRAL (my local CBS affiliate) on that date will be the winner.
And now....time to compare notes!
One of the great things about The Big Wheel is the fact that, in a way, it has built in probability. The numbers on the wheel correlate directly to some of the percentages that come up in the calculations. I used that to my advantage in my calculations. Now....I have taken plenty of math classes, but I don't have any official statistics training, so my notations may be a bit on the rough side, but I still believe they are right.
For the record...I did this in Excel
The 1st thing I did was create a formula to use the actual number on the first spin to represent the chance of winning, drawing or losing on subsequent spins. That number will remain consistent throughout the rest of the calculation since the player will not be able to go back and spin again at a later time (once he stays, that's it). The number of chances to win is refered to as x and is simply the spin total times 20 -1 or x = First_spin * 20 – 1
So the chance to win is x/20
The tie is always 1/20
and the chance to lose is (19-x)/20
I started out simply with a “head to head” solution. The scenario where the 2nd spinner has 'free will' to either spin or stay (if he trails the 1st spinner, he really has no choice but to spin) and player 1 is out of the picture either having busted or trailing spinner 2's first spin. The real question to consider is, which is greater...the chance of going over by spinning again or the chance of being beaten if you stand?
Calculating the chance of going over is easy it's simply whatever you spun. (ex: spin 0.50 = 50% chance 0.75 = 75% chance to go over, etc.)
Player 3's first spin is easy...just use the formulas above. It's the 2nd spin (or lack of) where things become tricky. Player 3 will not spin again on a win or ties in the cases we are considering. Here are the results so far:
Player 3 wins: (19-x)/20
Player 2/3 tie: 1/20 (5%)
Player 2 survives the 1st spin: x/20
We must then consider the 2nd spin. The only case of a 2nd spin in this case is if player 2 survives the 1st spin (as stated above player 3 would not spin on a tie above 0.50). In the case where player 3 does not spin again, you would simply take the formula on face value. For the values where player 3 does spin again, you have to multiply the 1st spin formula by the 2nd spin result. The result is:
Player 3 wins in one spin: (19-x)/20
Player 3 ties in one spin: 1/20
Player 2 survives 1st spin/loses on 2nd spin: x/20 * (19-x)/20 or (19x-x^2)/400
Player 2 survives 1st spin/ties on 2nd spin: x/20 * 1/20 or x/400
Player 2 wins: x^2/400
You then have to add all the loss and tie values (there is only one win value):
overall chance to:
loss: (380-20x)/400 + (19x-x^2)/400 reduces to (-x^2 – x + 380)/400
tie: 1/20 + x/400 or (x+20)/400
win: x^2/400
From this point on these formulas will be referred to as TL(total chance of loss), TT(total chance of tie), and TW(total chance to win). These win, loss and tie calculations will be used for my calculations for player 1, but for the purposes of calculating the odds for player 2 there is one more thing left to resolve. The tie. The spinoff is pretty much a coin flip....one spin winner take all. Therefore, you must split the tie result evenly between the wins and losses. Here were my results for the “borderline” spins for player 2:
0.70: 46.375% to win when standing vs 30% safe 2nd spin
0.65: 40% to win when standing vs 35% safe 2nd spin (← stand)
0.60: 34.125% to win when standing vs 40% safe 2nd spin (← spin again)
Using the same method you now must calculate Player 1's optimum decision. It can be calculated as 2 independent head to head matchups using the TW, TT and TL formulas...
Player 1 beats Player 2:
...but loses to player 3 (TW * TL) (loss)
...ties player 3 (TW * TT) (2 way tie)
...beats player 3 (TW * TW) (WIN)
Player 1 ties Player 2:
...but loses to Player 3 (TT * TL) (loss)
...ties player 3 (TT * TT) (3 way tie)
...beats Player 3 (TT * TW) (2 way tie)
Player 1 loses to Player 2 (player 3's result is irrelevant): TL (loss)
Just like in the 2 player scenario you should add all the loss and tie results (there is only one win result). For the Player 1 losing to Player 2 result, it would be necessary to multiply by 20/20 to match up denominators (or in my case, I just allowed all the percentages to be translated by the spreadsheet). Then split up the ties. The 2 way ties are split evenly between the wins and losses and the 3 way ties are split 1:2 between wins:losses respectively (Player 1 will only win 1/3 of all 3 way ties).
My results for the borderline spins for player 1:
0.75: 28.42% to win by standing vs 25% safe 2nd spin (← close but don't spin)
0.70: 21.56% to win by standing vs 30% safe 2nd spin (← spin here)
There is one more scenario to consider. Player 2 ties Player 1 on the 1st spin...
This was actually easier than I 1st expected. Simply use the same formulas from the Player 2 vs Player 3 section only now, you are treating all P2/P3 ties as 3 way ties and all wins as 2 way ties. Here are the borderline spins for Player 2 with the new tie splits factored in.
0.75: 27.33% to win by standing vs 25% safe 2nd spin (← very close, but stand)
0.70: 23.88% to win by standing vs 30% safe 2nd spin (← spin here)
discuss....
Current Mood: ![[mood icon]](http://l-stat.livejournal.com/img/mood/niaha/kitty/anksleepyg.gif) exhausted
|
|
|
|
|
|
LiveJournal.com |
![[info]](http://l-stat.livejournal.com/img/userinfo.gif){kind=small}
d_l_leonine's journal
