Ask The Wizard #366

To arrive at an answer, I began by analyzing how frequently each letter appears in every position, drawing from the data provided. list of allowed Wordle solutions .

Next, I examined the list of words that solve the Wordle puzzle, focusing on those with five unique letters and evaluated them based on the letter frequency chart. I assigned two points for letters that matched in the correct location and one point for those in incorrect positions. The final list is ranked below.

Consequently, my suggested starting word, which I personally prefer, is STARE.

Here is my solution (PDF).

First, let's determine the chance that the player will be down by more than x units after conducting one million flips, assuming they have an endless supply of money.

Given that this is a fair wager, the average outcome after a million flips is zero. Each flip has a variance of 1, resulting in a total variance of one million for one million flips. Therefore, one standard deviation equates to sqrt(1,000,000) = 1000.

We can compute the necessary bankroll using the Excel function =norm.inv(probability,mean,standard deviation). For instance, plugging in =norm.inv(.25,0,1000) yields -674.49. This indicates that after one million flips, the player has a 25% chance of being down 674 or more. It’s important to note that this is merely an estimate; for accuracy, we would need to apply the binomial distribution, which could be quite complex for a million flips.

There's a possibility that if the player brings $674 to the table, they might exhaust their funds before completing one million flips. If they have the option to continue playing on credit, they may even recover and finish with losses under $674. In fact, once the player reaches -674, there’s an equal chance, 50/50, of ending up above or below that threshold at any future moment.

Therefore, assuming the player is allowed to play on credit, there are three potential outcomes.

1. Player never falls below -674.
2. The player goes below -674 at some point but manages to recover and ends with losses above -674.
3. The player drops below -674 at some stage, continues to play, and incurs even more losses.

We've established that Scenario 3 has a 25% probability.

Scenario 2 should have an identical probability to Scenario 3 because, once the player hits -674, they have an equal chance of finishing above or below that figure after one million flips.

Scenario 1 represents the remaining option, which must therefore account for the probability of 100% - 25% - 25% = 50%.

If the likelihood that the player never dips below -674 is 50%, then the chances of dipping below must also be 100% - 50% = 50%.

Thus, we've answered the initial query: $674.

This topic has been raised and discussed within my forum at Wizard of Vegas .

This topic has been raised and discussed within my forum at Wizard of Vegas .