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Ask The Wizard #301
Every week, the nearby casino hosts a lottery where one lucky winner stands a chance to receive either $5,000 in cash or a mysterious bag filled with various prizes, including options such as:
- $2,000 cash
- $4,000 non-negotiable chips
- $6,000 non-negotiable chips
- $8,000 cash
- $10,000 cash
What would you choose and why?
Initially, I believe the 'non-negotiable chips' refer to those that are valid until they are fully utilized. Their worth is roughly 98% of their original value, contingent upon the specific game rules applicable.
On a related note, the average value of the mystery bag is around $5,960, which is significantly higher, nearly 20% more than taking the cash alternative. Even when considering the utility of money, opting for the bag is advantageous, particularly if you lack other financial resources.
This topic was brought up and examined in a discussion on my online forum at Wizard of Vegas .
What is the statistical expectation for the number of random draws from a uniform distribution between 0 and 1 before their total reaches 1?
What was the math problem on the chalkboard in the movie Good Will Hunting ?
Interestingly, solving this problem was fairly straightforward, particularly for a combinatorial mathematics course at MIT. Here is how the problem is stated:
\"Create all homeomorphically irreducible trees when n equals 10.\"
Let me simplify that for you.
Using only straight lines, create all designs such that the total count of intersections and dead ends equals 10. Closed loops are not permitted, and you cannot produce two identical designs. Each intersection must emanate from at least three distinct pathways.
What do I mean by 'identical' figures, you may wonder? It refers to the ability to rearrange the components while maintaining the integrity of the intersections, resulting in no new designs.
Here is an example:
Here's a clue: Contrary to the conclusion reached in the film, there are in fact ten distinct solutions. Will identified only eight. Try to either replicate or surpass Will Hunting's findings.
- MATHEMATICS IN GOOD WILL HUNTING II: CHALLENGES FROM THE STUDENT'S POINT OF VIEW -- Academic paper on the problem.
- THE GOOD WILL HUNTING MATH PROBLEM -- Analyzing the issue further in my forum.
In some casinos across Mexico, roulette is played using dice instead of the traditional wheel. Here’s how it works:
- The game consists of four dice—two green, one red, and one blue.
- When both green dice land on one, the resulting 'spin' will yield a score of zero.
- If both green dice show six, this will result in a double zero for the spin.
- For any other results with the green dice, the 36 possible outcomes from the red and blue dice will be assigned to represent the numbers 1 through 36 for the 'spin'.
How does this alter the betting odds compared to the standard roulette game?
The chances of hitting a 0 or 00 would sit at 1 in 36 each. If wagers on these outcomes paid out at the regular 35 to 1 ratio, the house edge would effectively be 0%.
The likelihood of any other number winning would be calculated as (34/36)*(1/36) = approximately 2.62%. This is in contrast to the 1/38=2.63% found in conventional double-zero roulette. The house edge for bets on numbers 1 to 36 would be 5.56%, while standard double-zero roulette sits at 5.26%. My suggestion for this game is to bet exclusively on the zero and double-zero.
If anyone can verify or contest these regulations and payouts, I would appreciate your input.