The Price is Blog, Right?

Recently, “A huge fan” commented on my strategy guide to Cover Up, and he and I continued discussing whether or not there might be a better strategy out there. I assume he means he’s “a huge fan” of TPIR, not of the blog, since it should already be assumed that he’s a huge fan of the blog.  Being a huge fan of TPIR is not necessarily a given.  I think it’s safe to assume that there are countless readers of this blog who may not be fans of TPIR at all.  Or they could just be medium-sized fans.

For convenience, I’ll refer to “A huge fan” as “huge fan” from now on.  Not: this type of shorthand will be used frequently in the blog.  To save time.  Wouldn’t want to waste space on unnecessary articles, or pointless explanations.

Huge fan had some pretty interesting ideas, and after he sent them to me, he actually decided that he made some unfair assumptions while doing the calculations, and thus it is not entirely accurate.  But I feel it’s thought-provoking and insightful, even if not 100% useful, like my theory.  It is extremely long, so please feel free to not read it.  The basic idea is that my strategy works.  Also that if you know more than one number for sure, you should spread them out across multiple rounds and not guess them all at once (obviously).  And that it’s almost impossible to run the statistics with every possible variation through to the end (but if anyone out that can do it, you’re welcome to re-guest blog and replace huge fan as the cover up semi-expert).  So here is an abridged version of huge fan’s idea:

For the sake of this post, let’s assume that you are 100% sure of the first digit, and have no clue about the other 4.  Obviously this is a hypothetical situation, but it probably isn’t that far away from reality (because the first digit only has 2 choices, and as PriceIsBlog says, it’s an extremely guessable digit. Regardless, this will help illustrate general strategy).  So the question becomes, when should you play that digit?  Let’s look at 2 possibilities, playing it first (the PriceIsBlog way), or playing it second (let’s call this the HugeFan way).

To understand which is better, we need to know our probability of guessing at least one digit correctly at each round.  Just for kicks, let’s take the entire game at the beginning, assuming we know nothing (not even the first digit).  What’s the probability of guessing at least 1 correct digit?  Well we know that the odds of each digit are 1/2, 1/3, 1/4, 1/5, 1/6 respectively.  So the trick with figuring out the probability of AT LEAST one, is to first figure out the probability of getting them all wrong, which is: (1/2)(2/3)(3/4)(4/5)(5/6) = 16.7%.  Notice that to calculate that, we use the probability of getting it wrong, so for example, for the last digit, it’s 5/6 of the time you’re wrong, not 1/6.  Okay, so the probability of getting at least one right is the inverse of that: 100% – 16.7% = 83.3%.  Not too bad?

Alright, now that we know how to do the math, back to our scenario.  So in PriceIsBlog’s situation, you guess the first digit right at the beginning.  For the sake of easy calculations, let’s say he nailed his strategy, and got every other digit wrong.  There are 4 numbers left, but their probability has changed.  The second digit originally had a 1/3 chance of getting it right, but he got a wrong answer, so in the second round, it’s now 1/2.  The third digit had a 1/4 probability, so now it moves to 1/3.  Same goes for the remaining digits.  So calculating our odds of getting AT LEAST one digit in the second round is now 1 – (1/2)(2/3)(3/4)(4/5) = 80%.  Finally, let’s say he guesses again (for the sake of calculation, gets the easiest number, but none of the others).  So he’s left with digits 3,4,5.  His probability at that point would then be (remembering to adjust the probs on each digit): 1-(1/2)(2/3)(3/4)=75%.  Alright, so there we have it.  To summarize, on the first guess, PriceIsBlog has 100% prob (he knows the answer), on the second, 80% and on the third 75%.

Now, let’s do the HugeFan strategy, saving that first digit.  The probability on the first round (on every number but the first digit) is: 1-(2/3)(3/4)(4/5)(5/6) = 67%.  Let’s assume once again that we guess the easiest number but none of the others.  On the second round, it’s 100% (we know that first digit), and finally, on the third round, we’re left with digits 3,4,5.  The probabilities of hitting those on the third round are 1/2, 1/3, 1/4, so the odds of getting at least one right are: 1-(1/2)(2/3)(3/4) = 75%.

PriceIsBlog: Rd 1: 100%, Rd 2: 80% Rd 3: 75%

HugeFan: Rd1: 67% Rd. 2: 100% Rd 3: 75%

Since making it part way doesn’t even get you a hubcap, clearly PriceIsBlog’s strategy is a better one.  So if you know the first digit, play it first.

Let’s step back a minute, to absorb what’s going on.  Each round that you are alive, your probability of guessing at least one digit right gets worse.  This is unavoidable, no matter what you guess (right or wrong).  However, these diminishing odds are partially offset by clearing out wrong answers (as PriceIsBlog mentioned), which raises some individual digits probabilities.  So, to utilize this information, you should spread out the digits that you think you know for sure, and don’t play them at once.  Let’s say you were 99% sure about digits 1 and 5.  Your odds only improve a tiny bit by guessing them at once, so you should clearly do it one at a time, and clear out some other wrong digits in the process.  Finally, by taking what we learned in the original hypothetical, make sure to play your first known digit at the start.

**It should be noted that these calculations had a lot of assumptions, and are quite possibly wrong.  Someone with a stronger probability background (like an actuary) feel free to edit/respond to this post.