Ask The Wizard #149

While I may not find Parrondo’s paradox particularly captivating, I have had several inquiries about it, so I’ll share my perspective. The core idea is that by switching back and forth between two specific losing games, a player can actually acquire an edge.

For instance, in Game 1, the chances of winning $1 are 49%, while the risk of losing $1 stands at 51%. In Game 2, when the player’s bankroll is divisible by 3, they have a mere 9% chance of winning $1, and a daunting 91% chance of a loss. Conversely, if the bankroll isn’t divisible by 3, the player has a much higher 74% chance of winning $1 and a 26% chance of losing.

It's evident that Game 1 produces an expected outcome of 49% multiplied by 1 plus 51% multiplied by -1, resulting in a -2% expected value.

In Game 2, you cannot simply calculate a straightforward average of the two scenarios because winning causes the game to quickly deviate from a bankroll remainder of 1, and it often shifts between remainders of 0 and 2. Essentially, this leads to a disproportionate likelihood of engaging with the game that has only a 9% winning chance. Thus, if you only play Game 2, the expected value lands at -1.74%.

By interchanging two rounds of Game 1 with two rounds of Game 2, we disrupt the alternating nature of Game 2. As a result, players end up playing the game with a 75% winning chance more frequently, while the game with a 9% winning chance becomes less prevalent. There are countless combinations to mix both games. Employing a strategy that alternates two games of each type gives an expected value of 0.48%.

It’s crucial to highlight that this notion has no practical application in a casino setting. Casino games do not alter their rules based on the modulo of a player’s bankroll. Nevertheless, I anticipate that it’s only a matter of time before someone markets a betting system based on Parrondo’s concept, by switching between roulette and craps, which will inevitably be just as ineffective as all other betting strategies.

Indeed, when looking at individual events, if the probability is p, the average wait time equals 1/p. However, this becomes more complex with consecutive events. Let x represent the condition where the last roll wasn't a two, which is also the initial state. Let y represent the condition where the last roll was a two. After the first roll, there’s a 5/6 chance we remain in state x, and a 1/6 chance of moving to state y. Let Ex(x) denote the expected number of rolls from state x, and Ex(y) from state y. Then..

Ex(x) = 1 + (5/6)*ex(x) + (1/6)*ex(y), and
Ex(y) = 1 + (5/6)*ex(x)

Solving for these two equations..

Ex(x) = 1 + (5/6)*ex(x) + (1/6)*( 1 + (5/6)*Ex(x))
Ex(x) = 7/6 + (35/36)*Ex(x)
(1/36)*Ex(x) = 7/6
Ex(x) = 36*(7/6) = 42

Thus, for two consecutive twos, the expected wait time remains at 42 rolls.

I’m facing a similar issue, only my query revolves around the expected number of flips to achieve two heads, in my case. math problems , see problem 128.

I believe that it’s nearly impossible for ex-partners to remain friends after a breakup. The most one can hope for is a distant relationship where they exchange holiday greetings. Any level of closeness implies at least one person is harboring hopes of reconciliation. While you didn’t ask for my opinion on breakups, I’d advise cutting ties cleanly to move forward with your life. I can’t say for sure what’s occurring here, but where there’s smoke, there’s fire. Although you lack concrete evidence to make accusations, it’s wise to remain vigilant.

Interestingly, I’ve heard of a casino in Vegas that’s taking similar actions regarding their Caribbean Stud game. Here’s a formula to compute the expected return when a straight flush pays the complete jackpot: (((5108*FL+3744*FH+624*FK+40*J)/2598960)-M*0.052243-1)/(M+1) where

FL = Flush win
FH = Full house win
FK = Four of a kind wi
n J = Jackpot amount
M = Minimum ante bet

In your instance, the calculation looks like this: (((5108*150+3744*300+624*1500+40*155000)/2598960)-5*0.052243-1)/(5+1) = 36.858%. This means the player’s advantage is 36.858% of the total ante plus the $1 side bet, resulting in an expected profit of $2.21 per hand.

The exact answer 1-(9,999,999/10,000,000)^10,000,000= 0.632121. This is the same as (e-1)/e when rounded to seven decimal places.

Such behavior would be deemed poor poker etiquette. If you attempted this in Vegas, you might receive a warning on the first instance, but a second violation could result in your removal from the table.

I firmly believe that the manufacturers of shuffling machines strive to make them as fair and random as possible. If a machine is deliberately tampered with, it would likely breach Nevada regulations. It’s relatively common to observe strong x-card hands when dealt x+1 cards. For instance, the probability of obtaining three of a kind with three cards is just 0.235%, whereas it rises to 0.922% when four cards are involved, making it almost four times more probable.

Firstly, I want to express that I am supportive of dating coworkers! I do not adhere to rules that prohibit interoffice relationships. Meeting new people can be challenging enough without limiting your interactions with familiar faces. Nevertheless, I would suggest caution if both parties fall under the same chain of command. It doesn't sound like this is the case, so I wouldn’t let that hinder your pursuits.

The fact that you have established a friendship is a great indicator. I know this may sound immature, but could you enlist the help of someone you trust to gauge her feelings about you? Alternatively, you could arrange a casual dinner after work with her and some colleagues. In a more relaxed setting, possibly with a few drinks, you could subtly explore her sentiments without outright professing your emotions. It’s crucial to raise the temperature of the situation without risking your friendship by declaring your feelings outright.

In response to your question, the likelihood that you are developing feelings for her is, without a doubt, nearing 100%. As for her feelings, it’s more ambiguous—probably around 10%. Nevertheless, don’t let this deter you; she might just need more time to understand her own emotions. Wishing you the best of luck! I would love to hear how things unfold.