Probability of the Price is Right

[Rusty]

I've seen contestants do this.  That turns out to be the correct answer.  So no, the probability is still 25% therein.

Cyclone

After a contestant eliminates the correct answer, the probability of success is still 25%?

I should have added the critical phrase "provided they didn't eliminate the answer."  Then the probability is a big fat 0.

[Cyclone]

After a contestant eliminates the correct answer, the probability of success is still 25%?

I should have added the critical phrase "provided they didn't eliminate the answer."  Then the probability is a big fat 0.

I'm not being clear.  Whether a contestant eliminates an answer beforehand or not, the probability of a correct answer is still 25%.  A contestant can eliminate an answer, but there is still a 25% probability that the answer could be correct.  It's about randomness, just like rolling a die.  On any die roll of a single die, you have a 1/6 chance - roughly a 16.67% chance - of rolling any one of the six digits from 1-6.  When you want to roll the same number twice consecutively, you still have a 1/6 chance after the first roll is already decided to get the same number.  If you call for a number (say, five), you have a 1/6 * 1/6 chance of rolling the same number (five) twice in a row - basically a 2.8% chance.

Like looking at the example with a die, using any five random digits, we've determined that there are 120 different possible combinations, not counting any special rules.  When one digit is a 0, that eliminates an option for the first number by default, but otherwise 120 is accurate.  Therefore, the probability - I stress, probability - of getting the correct price within 8 tries (since two have to be used by the other prizes) with five random digits is 8/120 = 6.67%.  When you factor in special rules or pricing strategy, it doesn't automatically make it correct; probability knows nothing of that nature.

Math was my strong point growing up, this is something I can discuss in detail.

Cyclone

[stardf29]

Would someone explain to me where I went wrong with my calculations?  As I mentioned before, I don't have Inspiration 7.6 or higher, so I can't open the file.

(I think the mistake might have to do with having not factored in the fact that contestants only get ten chances.)

Assuming the 0 rule:
First prize: 1/2 chance of getting it right
Second prize: 1/6 chance of getting it right
Third prize: 1/24 chance of getting it right

1/2 + 1/6 + 1/24 = 17/24 = 70.83%

Not Assuming the 0 rule:
First prize: 1/6 chance of getting it right
Second prize: 1/24 chance of getting it right
Third prize: 1/120 chance of getting it right

1/6 + 1/24 + 1/120 = 26/120 = 21.67%

10 Chances is a bit different in that it is something more like this:

Assuming the zero rule, for the first prize, it is like rolling a 2-sided die (well, flipping a coin, but the coin has one side with "1" and one side with "2"), then the second prize is like rolling a 6-sided die, and the car is like rolling a 24-sided die, the sum of what is rolled is added up and the question is, what is the probability that the sum is 10 or less?

Similarly, not assuming the zero rule, it is like rolling a d6, d24, and a d120 and adding them up and seeing if it is 10 or less.

The probabilities are going to be a bit weird, with how this all works out. Obviously, if we do not assume the zero rule, the probability is going to be very, very low for winning...

[Rusty]

I'm not being clear.  Whether a contestant eliminates an answer beforehand or not, the probability of a correct answer is still 25%.  A contestant can eliminate an answer, but there is still a 25% probability that the answer could be correct.  It's about randomness, just like rolling a die.  On any die roll of a single die, you have a 1/6 chance - roughly a 16.67% chance - of rolling any one of the six digits from 1-6.  When you want to roll the same number twice consecutively, you still have a 1/6 chance after the first roll is already decided to get the same number.  If you call for a number (say, five), you have a 1/6 * 1/6 chance of rolling the same number (five) twice in a row - basically a 2.8% chance.

Like looking at the example with a die, using any five random digits, we've determined that there are 120 different possible combinations, not counting any special rules.  When one digit is a 0, that eliminates an option for the first number by default, but otherwise 120 is accurate.  Therefore, the probability - I stress, probability - of getting the correct price within 8 tries (since two have to be used by the other prizes) with five random digits is 8/120 = 6.67%.  When you factor in special rules or pricing strategy, it doesn't automatically make it correct; probability knows nothing of that nature.

Math was my strong point growing up, this is something I can discuss in detail.

Cyclone

I have a masters in math and math education, and math was my strong point growing up too.    Saying things like "we know the price will end in 0" or "the first digit of the price of the car can't be 6 or 8" changes the probability of success since it reduces the size of the sample space.  If such things are ignored and the contestant is blindly, randomly picking digits, then yes, the full sample space of 5! = 120 prices (even allowing for a leading 0) are in play.  Even still, your probability is wrong, since after the first guess there are only 119 prices to choose from, then 118, etc. (duplicate prices aren't allowed by rule). 

As for the Millionaire scenario, eliminating the correct answer and then randomly selecting one of the three remaining (incorrect) answers is effectively removing the only successful outcome from your sample space.  Your probability goes from 1/4 to (1-1)/(4-1) = 0/3 = 0.

Edit: I'll be damned.  After setting pencil to paper (and letting P(x) be the probability it takes x chances), I see that P(1) = 1/120, P(2) = (119/120)(1/119) = 1/120, P(3) = (119/120)(118/119)(1/118) = 1/120, ..., P(8 ) = (119/120)(118/119)...(113/114)(1/113) = 1/120; since the probabilities are mutually exclusive, we add 'em up to get 8/120.  Apologies for jumping the gun, Cyclone.

[SteveGavazzi]

Even still, your probability is wrong, since after the first guess there are only 119 prices to choose from, then 118, etc. (duplicate prices aren't allowed by rule).

I've lost track...are we talking about Ten Chances?  Because if we are, duplicate guesses are allowed and have occasionally gotten through...it generally depended on whether or not Bob noticed it before he hit the button.

(Amusingly, this is contradicted by a portion of the rules which states that not winning the first prize is impossible.)

[Cyclone]

Apology above noted.

I do concede the point on Millionaire that eliminating the correct option reduces your chances of winning to a goose egg.  However, the basic probability is still a 1/4 scenario if there is no assumption that the correct answer is being removed.  In that sense, you can remove any answer and you still have a 25% chance in the initial probability that what you eventually choose is correct.

Let's go into a specific example using the Millionaire scenario.

Q: How many Americans have walked on the moon?

A: 6
B: 9
C: 10
D: 12

You are presented with four answers.  The basic probability is 1/4.  I however know the answer to this question.  Yes, that means that my personal probability of correctly answering this question is 100%.  Basic probability, however, still assumes a sample size including all four options.  Some people - possibly even folks here - do not know the answer to this question.  Removing items from the sample does not reduce the basic probability; Steve might think the answer is A, while vader might suggest B, and JokerFan might say C while Rusty says D.  All four have a 25% probability of being correct.  And anyone reading the question might be 100% certain of the answer, eliminate all other options...and still be wrong.  There is a classic Millionaire moment when a Phone-A-Friend does just that, getting all excited about a question and yelling the answer over and over...and it's the wrong answer.  Both thought they had it 100% correct.  Both were wrong.  The 25% probability held true, and what we are exploring in Ten Chances and all other games is just that, basic probability.  It's all right to make assumptions, but they do not ruin the basic probability, which is what I explore in my earlier posts.

Cyclone

[JohnHolder]

You're wrong. 

You're right.

A contestant with certain information (e.g., knowing that the prize ends in 0 and doesn't begin with 6) has a higher probability of winning than a contestant who doesn't have that information. I get that and I wasn't trying to argue that with vadernader (even if he seems to think otherwise). I was trying and failing to make a distinction between the effect of contestant knowledge and a formal change in the parameters of the game: e.g., the rules of the game don't explicitly state that the price must end in 0, therefore a non-zero choice can't be eliminated from the formal probability of winning even if anyone who watches the game more than once can figure out that non-zero actually isn't correct. There's a different statistical probability of success for a game where the host says explicitly, "The first price is either 50 or 70" as opposed to saying, "Form the price using two of the three numbers 5, 7 and 0 [wink, wink, nod, nod]." That's the point I was trying to make, and I didn't make it well.

We can calculate the probability given that the contestant knows the Zero Rule as opposed to the probability if he doesn't know it, or the probability given (or not) that the player knows that a Nissan Versa isn't $60K, but I think I got tripped up by the fact that we can't know the probability that the player knows those things until he actually starts playing, and we can't actually determine whether he knows the Zero Rule until he makes a bid of something like 57, in which case we know that the probability is 100% that he doesn't.

[Rusty]

<snip>

But your definition of 'basic probability' only applies to blind, random events.  Yes, if I close my eyes and plug my ears, wait until I think Meredith is done talking, then say "C, final answer," I have a 1/4 shot of being correct.  If I worked at NASA or watched a moon documentary on PBS the night before, it tips the scales in my favor.  Being able to eliminate incorrect answers does enhance your probability.  When taking standardized tests, high school students are encouraged to eliminate incorrect answers and guess when they don't outwardly know the answer.  There's a reason for that.

[Rusty]

You're right.

OK, you've got my attention.  Please continue.

<technical, non-flattering stuff snipped>

Agreed on your point.  Additional information (zero rule, no $60K cars) reduces the number of possibilities which increases the probability of success.

[Cyclone]

But your definition of 'basic probability' only applies to blind, random events.  Yes, if I close my eyes and plug my ears, wait until I think Meredith is done talking, then say "C, final answer," I have a 1/4 shot of being correct.  If I worked at NASA or watched a moon documentary on PBS the night before, it tips the scales in my favor.  Being able to eliminate incorrect answers does enhance your probability.  When taking standardized tests, high school students are encouraged to eliminate incorrect answers and guess when they don't outwardly know the answer.  There's a reason for that.

Just because an answer is eliminated doesn't mean the probability of being correct increases.  Do you know how many multiple choice questions I've answered incorrectly?  I eliminated the correct answer and chose something else.  If what you say is correct, then by eliminating three answers I have a 100% probability of a correct answer on every question.  It doesn't work out that way 100% of the time.

Cyclone

[JohnHolder]

by eliminating three answers I have a 100% probability of a correct answer on every question.  It doesn't work out that way 100% of the time.

You have to know which three answers to eliminate.

[Cyclone]

You have to know which three answers to eliminate.

Exactly.

And we can't assume knowledge in basic probability.  On other points, I do agree that your chances improve if you know an answer that is incorrect.

Cyclone

[PriceFanArmadillo]

Um...okay, at this point, even I can't tell if we're on topic any more. 

It looks like we're in the midst of a Monty Hall scenario here, and as best as I can tell, Barker's Markers hasn't been the topic of conversation lately.  Enough with the multiple-choice questions; I don't think a Millionaire-style question has ever been asked on The Price is Right. 

[bduddy]

Um...okay, at this point, even I can't tell if we're on topic any more. 

It looks like we're in the midst of a Monty Hall scenario here, and as best as I can tell, Barker's Markers hasn't been the topic of conversation lately.  Enough with the multiple-choice questions; I don't think a Millionaire-style question has ever been asked on The Price is Right. 

Almost every discussion on the Internet about the probability of anything eventually comes down to this: a discussion on the nature of probability itself. Don't worry, it'll peter out eventually...

[Rusty]

If what you say is correct, then by eliminating three answers I have a 100% probability of a correct answer on every question.

No, if you correctly eliminate 3 incorrect answers, you're guaranteed a win. 

Outside of this conversation I've never heard the phrase 'basic probability,' and I've got a feeling you've never heard of 'conditional probability.'

Almost every discussion on the Internet about the probability of anything eventually comes down to this: a discussion on the nature of probability itself. Don't worry, it'll peter out eventually...

So true.  Probability theory has only been around since the late 18th century, and many results are counter-intuitive.  (The Monty Hall problem comes immediately to mind.)