Joeman
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February 21st, 2014 at 6:15:13 PM permalink
Several years back, maybe in 2008 or so, the Wheel of Fortune game show added a $1 million wedge to the wheel. The rules, as I understand them, state that for a contestant to win the $1 million, he must accomplish all of the following:

1) In the main game, spin the wheel such that the wheel land exactly on the center 1/3 of the wedge (the outer 1/3's are Bankrupt spaces). I read that there are 24 wedges on the big wheel.
2) Call a letter that is found in the puzzle. (The contestant picks up the wedge at this point.)
3) Solve the puzzle for the round in which he obtains the wedge.
4) Be the Big Winner of the main game. (Without hitting a Bankrupt from the point at which he picks up the wedge)
5) Spin the bonus prize wheel such that the the wheel lands on the $1 million prize. (Which replaces the standard $100,000 prize on the bonus wheel if #1-4 above have been successfully completed.) I read that there are also 24 spaces on the bonus wheel.
6) Solve the bonus puzzle.

The contestants all seem to go gaga about the $1 million wedge if they have a chance to get it, and the host seems to encourage this fervor. However, I am skeptical that the addition of this wedge to the wheel increases the contestants' EV at all. In addition to the long odds of wining the $1 million prize, the addition of the 2 Bankrupt sections makes me think the addition of this wedge (which I believe replaced a dollar amount wedge) is overall bad for the contestants. But I have obviously not done the math.

So my questions are, did the addition of the $1 million wedge increase the EV for the contestants? If so (or even if not!), if the $1 million wedge were replaced by a standard money amount wedge, what would that amount be that would offer the same expected value to the contestants?
"Dealer has 'rock'... Pay 'paper!'"
Mosca
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February 21st, 2014 at 6:25:14 PM permalink
I don't know the answer, but two people have won the $1,000,000.
A falling knife has no handle.
endermike
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February 21st, 2014 at 6:31:10 PM permalink
Cool question. I will give it a try. One thing we should try to establish up front is are we interested in how it effects an average player, or a "good player." When I say a "good player" I mean one who is above average at the game and more likely to be able to solve these puzzles generally and than their opponents.

Which do you think we should use?
endermike
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February 21st, 2014 at 6:32:33 PM permalink
Quote: Mosca

I don't know the answer, but two people have won the $1,000,000.



This is helpful. Does anyone know how many-ish episodes they have had this feature? Further, does anyone have a good guess how many players get to other stages of the process?
Ibeatyouraces
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February 21st, 2014 at 6:41:37 PM permalink
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DUHHIIIIIIIII HEARD THAT!
beachbumbabs
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February 21st, 2014 at 6:52:56 PM permalink
Here's a first piece of info:

million

The million dollar wedge premiered on Sept 8, 2008, but

1. it was a full wedge, not a part-bankrupt.
2. it was initially only on the wheel for 1 round.
If the House lost every hand, they wouldn't deal the game.
beachbumbabs
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February 21st, 2014 at 7:02:59 PM permalink
There is another rule needed in the OP's list, which I found elsewhere;

The person holding the wedge must NEVER land on a bankrupt the rest of the game, even after winning the round for the wedge. I think any carryover (such as the wild card and the 1/2 car) is vulnerable to this rule as well, but I don't know that.
If the House lost every hand, they wouldn't deal the game.
Ibeatyouraces
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February 21st, 2014 at 7:05:10 PM permalink
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beachbumbabs
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February 21st, 2014 at 7:28:32 PM permalink
The wedge, in the round it's used, is placed on top of a 500 yellow wedge, which goes back into play if it's won, so there is an extra 2/3 wedge bankrupt on the wheel for all or part of that round only. Also from wiki/wheeloffortunehistory:
Several contestants have been only one or two pegs away from the $1,000,000 envelope. Two have lost the wedge to a Bankrupt, then lost $100,000 in the Bonus Round, while one won the $100,000 after losing the wedge. Also, one team during a Family Week won the $100,000 after failing to solve the round in which they claimed the wedge.
If the House lost every hand, they wouldn't deal the game.
Joeman
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February 21st, 2014 at 8:43:13 PM permalink
Quote: endermike

Cool question. I will give it a try. One thing we should try to establish up front is are we interested in how it effects an average player, or a "good player." When I say a "good player" I mean one who is above average at the game and more likely to be able to solve these puzzles generally and than their opponents.

Which do you think we should use?



Thanks for your interest, endermike. My hypothesis and the reason for my original question is that I think the $1 million wedge reduces the EV for all contestants. For that reason, I think we should figure it for the "average" contestant. I actually hope I am wrong because I hate to think that WoF would add a feature that actually reduces how much the players win.
"Dealer has 'rock'... Pay 'paper!'"
98Clubs
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February 22nd, 2014 at 2:30:29 PM permalink
1/24*1/72 for the wheels
I'll guess 1/5 for the letter (this could be much easier if partially revealed, and difficult at start, but with RNSTLE, could be easy)
Be the big winner... dunno how about 1/3
Don't ever go bankrupt... 1/5?
Solve the Bonus Round Puzzle... 1/15?

Thats 1 in 388,800
Some people need to reimagine their thinking.
Joeman
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February 24th, 2014 at 9:35:15 AM permalink
Thanks for the analysis, 98clubs. I agree that it certainly is a long shot to win the $1 million. However, I'm trying to approach it from a different angle. I'm wondering if the existence for the wedge on the wheel is overall good or bad for the contestants, and to what extent (+ or - expected win value).

Also, I would suggest that some of your probability assumptions might need a little tweaking. Based on the shows I have watched (I would consider myself a casual fan of the show), I think the probability of guessing a letter in a given puzzle would be closer to 1/2, and the Bonus Puzzle seems to get solved about every third time. This is strictly anecdotal observation. I am sure there is historical data out there somewhere that can give us more precise values... not nearly so sure it is available to us mortals.

I would also venture to say that the probability of both wining the round and winning the game are better than 1/3 since 1) If you pick up the wedge, it means there is already at least 1 letter revealed in the puzzle, AND it's your turn -- both factors that I would think increase your probability of solving; and 2) once you solve the puzzle in which you obtained the wedge, you will have banked more money than your opponents (on average), so your chances of an overall win are that much better. I could guess at these numbers, but I would bet there are those of you reading this that could make a better guess. And, as I hinted at above, someone with some historical data to go on could make an even better guess.

BTW, wouldn't the probability of NOT hitting the Bankrupt after picking up the wedge be 1 - (11/12) (n/2) where n is the total average # of spins per contestant (assuming the wedge is available in each round)? I bet we could get pretty close with that one.
"Dealer has 'rock'... Pay 'paper!'"
Joeman
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February 24th, 2014 at 9:37:15 AM permalink
Oof... I just saw on Wikipedia that the $1 million is paid out as a 20 year annuity. That's pretty crummy. C'mon WoF!
"Dealer has 'rock'... Pay 'paper!'"
slyther
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February 24th, 2014 at 3:42:25 PM permalink
Quote: Joeman

Oof... I just saw on Wikipedia that the $1 million is paid out as a 20 year annuity. That's pretty crummy. C'mon WoF!




That's better than "America's Got Talent" paying out their $1 million as a 40 year annuity!
Ibeatyouraces
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February 24th, 2014 at 3:51:22 PM permalink
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Joeman
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July 10th, 2014 at 11:00:32 AM permalink
I was watching WoF the other night, and someone "almost" won the million -- they solved the Bonus Puzzle, and were 2 spots away from the $1M envelope on the bonus wheel. This got me thinking about the value of the $1M wedge again. So, I decided to resurrect this thread and take a crack at it myself.

So, I came across the WoO site several years ago and really enjoyed the content. This past February, after returning from a Biloxi trip, I looked up the WoO site again and started reading some of the "Ask the Wizard" columns. After reading a few Q & A's about game shows, it brought to mind this topic that has been bugging me for some time.

Ever since I learned what it takes to win the $1M, I thought that it was not nearly as good a deal for the contestants as it was made out to be, though I never sat down and did the math. I can be lazy like that! :) I figured I could submit an "Ask the Wizard" question about it, and maybe The Great and Powerful One would decide to answer it. It would seem that he has a like for game shows.

Unfortunately for me, I noticed that he does not take/answer ATW questions any more, directing such queries to the Q&A section of the WoV site. So, I posted the question here. Although, hanging around here for the past fer months, I have learned a few things... like that when your very first post starts a thread, it may not get much attention. :)

Anyway, after I thought about it, it doesn't seem like the math is that hard (not to guarantee that I will get it right). However, there are several assumptions I have to make since I don't have any statistical data, and it does seem that the validity of an answer will depend heavily on the accuracy of these assumptions. So, here goes...


My original questions were: 1) Did the addition of the $1 million wedge increase the expected overall win amount (which I'll call EV) for the contestants? and 2) If the $1 million wedge were replaced by a standard money amount wedge, what would that amount be that would offer the same expected value to the contestants?

I'll put my answer under a spoiler tag. Feel free to let me know if you find any errors. (Not that I need to tell you guys that!)

1) No! The contestants would, on average win more if the $1M wedge was not on the wheel.
2) To get the same EV, you could replace the $1M wedge with a -$900 wedge!!! (Yes, that's negative nine hundred.)


To recap, the following 7 conditions must be met to win the $1M prize:
1) Spin the wheel to land on the center 1/3 of the $1M wedge.
2) Correctly guess a letter in the puzzle immediately after #1 above.
3) Solve the puzzle in which you obtained the wedge.
4) Avoid hitting a Bankrupt space for the remainder of the game.
5) Be the "Big Winner" at the end of the show.
6) Spin the Bonus Wheel such that the $1M envelope is selected.
7) Solve the Bonus Puzzle.

I'm not looking for the probability of hitting/winning the $1M; just the value of that particular wedge. So, I won't be including the probability of condition #1 in my calculations. All of the other conditions, except #6, will require either statistical data (which I don't have), or assumptions, which I will I will make below:

Probability Assumptions
Guessing a Letter: 0.5 -- Strictly a guess based on what I have seen
Solving the Puzzle: 0.4 -- A random event between 3 contestants would be 1/3, but if you gain the wedge, you have an additional letter(s) revealed in the puzzle AND it's your turn, both of these will increase your chances of solving.
Being the Big Winner: 0.35 -- Slightly above 1/3 since you must have won the round where you gained the wedge, banking at lease some money.
Solving the Bonus Puzzle: 0.33 -- Strictly a guess based on what I have seen

Other Assumptions
Avg spins per player per show: 12 -- Another guess, to be used in the Avoiding Bankrupt probability below
Average loss by hitting Bankrupt: $1,500 -- Big time guess. Remember, expected loss comes from both losing the money in front of you, and losing your turn.
Annuity Interest Rate: 5% -- Seems reasonable

Other Probabilities
Spinning the $1M envelope in the bonus round = 1/24 = 0.0417
Avoiding Bankrupt (2 Bankrupts, 24 Wedges, avg of 6 spins remaining) = (11/12)^6 = 0.593

So, to figure the value of the wedge, I must average the values of each third. 2 sections are Bankrupts, which I assumed to be valued at -$1,500. The center third is worth the value of the "$1M" prize multiplied by the probability of each of the conditions 2-7.

First off, the "$1M" prize is worth considerably less than $1,000,000 -- it is a 20-year annuity, plus it replaces the $100,000 prize that the contestant would get if there were no $1M prize:
50,000 * (1 - (1 / 1.05 ^ 20)) - 100,000 = $523,000

So, the value of the center third of the wedge is:

523,000 * .5 * .4 * .593 * .35 * .0417 * .33 = $299

The total value of the wedge = 299 - 2 * 1500 = -$900


First off, let me apologize... this post turned out to be way long. Second, WFT, Wheel of Fortune??!! Add a feature that hurts the contestants, yet tout it as a good thing? Shame, shame, shame!!! I might expect this type of nonsense from a casino (6:5 bj, anyone?), but not from you!

I've played with my assumptions on the spreadsheet, and no reasonable tweak can make this +EV. Even if it was the old way where the entire wedge was the $1M with no bankrupts, it's still worth less than the $500 space it covers up! Shame, WoF!
"Dealer has 'rock'... Pay 'paper!'"
Wizard
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July 9th, 2020 at 11:11:13 AM permalink
This question got asked of me recently for my live stream. It's a good question. Normally, I love game show questions, but Wheel of Fortune has never been a favorite of mine. To be honest, I stink at word games and that show makes me feel stupid.

Nevertheless, I'd like to take a stab at it. Here are some questions that will help:

1. Since this rule was introduced, how many games have there been and how often has anyone won the million? How often has anyone at least got to step 5?
2. Did this new wedge replace an old wedge, or is it an extra wedge?
3. What would you estimate is the probability of solving the final puzzle. In the example video below, the player didn't get it. I didn't either.
4. I know stupid question -- If the wheel lands on bankrupt, the player lose money from previous rounds or just the current round?

Bonus Prize Wheel

Direct: https://www.youtube.com/watch?v=KwSHBVkgOd4
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Mission146
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July 9th, 2020 at 11:42:23 AM permalink
Quote: Wizard

This question got asked of me recently for my live stream. It's a good question. Normally, I love game show questions, but Wheel of Fortune has never been a favorite of mine. To be honest, I stink at word games and that show makes me feel stupid.

Nevertheless, I'd like to take a stab at it. Here are some questions that will help:

1. Since this rule was introduced, how many games have there been and how often has anyone won the million? How often has anyone at least got to step 5?
2. Did this new wedge replace an old wedge, or is it an extra wedge?
3. What would you estimate is the probability of solving the final puzzle. In the example video below, the player didn't get it. I didn't either.
4. I know stupid question -- If the wheel lands on bankrupt, the player lose money from previous rounds or just the current round?

Bonus Prize Wheel


Direct: https://www.youtube.com/watch?v=KwSHBVkgOd4



I was able to get that one right, but I think I would miss probably as many as I would get, in general. You just kind of piece it together using the general rules of words. For example, second letter P in the first word makes the first letter a vowel almost certainly because S has been eliminated as a possibility...so now you just think of vowel-P words that end in double L. That process goes quickly because E and A have already been eliminated. There are very few words that start with, "IP," so you're mainly going to be looking at OP or UP words ending in a double L. No OP words that fit that characteristic even sound plentiful, so moving on to UP, I arrive at, "Uphill," fairly quickly. It also helps to immediately realize that the word cannot have five consecutive consonants so you have U-P-?-?-L-L that one of the ?'s must be a vowel.

The second word you now have three letters and, perhaps most importantly, know, "CL," which should be a word that associates with, "Uphill." That makes getting the second word trivial.

I could have just as easily missed and am probably no better at this than the average person, but such was my thought process in solving that particular puzzle successfully.

It's also difficult to determine whether you would have succeeded or failed based on what you have to try to do, primarily because you may not have guessed the same letters she did. C-M-P-A were quite reasonable letters to guess, though.

1.) Don't know, haven't watched in years.

2.) Don't know, but my guess would be replaced if it otherwise looks the same. (Rewatched: Sajak says they replaced $100,000 with $1,000,000)

3.) From when I used to watch regularly, and this is just throwing a number out there, maybe 25% solve the final puzzle.

4.) Current round only. Won money always carries over and all players get all money won in a specific round at the end of the game, even if they lose. So, if you win Round 2 but lose the game, you keep your Round 2 money and are paid that.
https://wizardofvegas.com/forum/off-topic/gripes/11182-pet-peeves/120/#post815219
billryan
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July 9th, 2020 at 11:42:41 AM permalink
These games are designed and retooled for ratings. Creating the illusion that someone might win a million is more important than anyone actually winning it.
The difference between fiction and reality is that fiction is supposed to make sense.
Mission146
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July 9th, 2020 at 11:53:17 AM permalink
If this article is correct:

https://newrepublic.com/article/116732/wheel-fortune-strategy-how-win-gameshow

As of 2014, 37.76% of contestants solve the final puzzle.
https://wizardofvegas.com/forum/off-topic/gripes/11182-pet-peeves/120/#post815219
ThatDonGuy
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July 9th, 2020 at 12:12:24 PM permalink
Quote: Wizard

This question got asked of me recently for my live stream. It's a good question. Normally, I love game show questions, but Wheel of Fortune has never been a favorite of mine. To be honest, I stink at word games and that show makes me feel stupid.

Nevertheless, I'd like to take a stab at it. Here are some questions that will help:

1. Since this rule was introduced, how many games have there been and how often has anyone won the million? How often has anyone at least got to step 5?
2. Did this new wedge replace an old wedge, or is it an extra wedge?
3. What would you estimate is the probability of solving the final puzzle. In the example video below, the player didn't get it. I didn't either.
4. I know stupid question -- If the wheel lands on bankrupt, the player lose money from previous rounds or just the current round?



#1 - the Million Dollar Wedge first appeared in the fall of 2008. I think there have been six players who spun the Million Dollar prize on the bonus wheel, and three of them won the million, but I'm not entirely sure.

#2 - as far as I know, the wheels have never changed size. I am not sure if the million/Bankrupt wedge replaced an existing Bankrupt space or not.

#4 - just the current round, although you have to remember that once a puzzle is solved, any money the other two players have in front of them is lost. There are some countries where the money does carry over from one round to the next, but that has never been the rule in the USA unless you include the old "On Account" rule (back in the days when a puzzle solver spent the money on prizes, some or all of the money could be put "on account" and used if they solved another puzzle later in the show, but they lost the money if they hit Bankrupt).
Joeman
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July 9th, 2020 at 1:12:59 PM permalink
I appreciate you taking a whack at this one, Wizard!

Quote: Wizard

1. Since this rule was introduced, how many games have there been and how often has anyone won the million? How often has anyone at least got to step 5?
2. Did this new wedge replace an old wedge, or is it an extra wedge?
3. What would you estimate is the probability of solving the final puzzle. In the example video below, the player didn't get it. I didn't either.
4. I know stupid question -- If the wheel lands on bankrupt, the player lose money from previous rounds or just the current round?

1) First off, I think we should discount the 2008 season, since when they introduced the $1 Mil wedge, the whole wedge was $1 Mil, not the 1/3 wedge with the adjacent Bankrupts that you see now. The $1 Mil prize was won by a contestant in October 2008, and not long after, they changed the wedge to its current configuration. Since then, 2 contestants have won the $1 Mil prize.

So, not counting 2008, there have been 11 seasons with 195 episodes per season for a total of 2145 shows (I'm not sure, but I believe the 2019-2020 season is over).

2) I'm pretty sure the $1 Mil wedge replaced an existing dollar amount. Currently, there is a $500 wedge under the $1 Mil wedge. So, once someone picks up the $1 Mil wedge, that space is now a common money space worth $500 per letter for the remainder of the show. Note that if a player gains the wedge, he earns no money on that particular turn, even if there are multiple occurrences of the letter he called in the puzzle.

3) Based on the frequency I have seen it solved, I would estimate 1/3.

4) As Mission & TDG pointed out, you only lose the money in front of you for the current round. Money and prizes accumulated from previous puzzles are not affected. However, if a contestant has obtained the $1 Mil wedge, and at any point afterwards spins a Bankrupt, he will always lose it, even if it was earned it in a previous round.
Last edited by: Joeman on Jul 9, 2020
"Dealer has 'rock'... Pay 'paper!'"
Deucekies
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July 11th, 2020 at 4:51:44 PM permalink
Quote: Joeman

I appreciate you taking a whack at this one, Wizard!

1) First off, I think we should discount the 2008 season, since when they introduced the $1 Mil wedge, the whole wedge was $1 Mil, not the 1/3 wedge with the adjacent Bankrupts that you see now.



This is false. The Million Dollar Wedge has always been a 1/3 wedge with bankrupts. When it is obtained by a player, they flip it over to show a full-size Million Dollar graphic, which may be what you're thinking of.
Casinos are not your friends, they want your money. But so does Disneyland. And there is no chance in hell that you will go to Disneyland and come back with more money than you went with. - AxelWolf and Mickeycrimm
Joeman
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July 13th, 2020 at 6:08:49 AM permalink
Quote: Deucekies

Quote: Joeman

I appreciate you taking a whack at this one, Wizard!

1) First off, I think we should discount the 2008 season, since when they introduced the $1 Mil wedge, the whole wedge was $1 Mil, not the 1/3 wedge with the adjacent Bankrupts that you see now.



This is false. The Million Dollar Wedge has always been a 1/3 wedge with bankrupts. When it is obtained by a player, they flip it over to show a full-size Million Dollar graphic, which may be what you're thinking of.

Wow, I must admit that I didn't watch much wheel in back in 2008, and I have forgotten who I heard this from, but I guess I got some bad info! Thanks!

ETA, I think this might be where I got the idea of it starting off as a full wedge.
"Dealer has 'rock'... Pay 'paper!'"
beachbumbabs
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July 13th, 2020 at 6:27:16 AM permalink
To further narrow down 4:

Quote:

4. I know stupid question -- If the wheel lands on bankrupt, the player lose money from previous rounds or just the current round?



A player wins whatever cash and prizes in each round, and they are safe from losing them, with several exceptions. 2 current ones: if a player has a wild card, or if a player has the million wedge, and they bankrupt, they lose those, even if they won them in a previous round.

Recent past: a player who had a half car in their bank would lose it in a bankrupt (or even a full car but hadn't yet solved the puzzle), because they could win the car over the course of the first 3 rounds, if they collected 2 halves. If the car was not won by the end of the 3rd round, all halves were forfeited. However, this season (37, just ended) they are not putting cars on the wheel - only in the bonus round.
If the House lost every hand, they wouldn't deal the game.
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