Ask The Wizard #192

Not long ago, I visited Foxwoods and observed the final stages of the Foxwoods Poker Classic. During my stay, Vince Van Patten, one of the hosts from the World Poker Tour, arrived and began to place various prop bets with some professional poker players nearby. He proposed a bet of 20 to 1 to anyone who could flip an entire deck of cards, naming each rank—Ace, 2, 3, and continuing up to King, then back to Ace—without hitting the card currently being announced. Despite the challenge, no one succeeded, and Vince pocketed several hundred dollars in about 10 minutes before the challenge was called off. While I think this trick is feasible, I can't help but wonder if Vince has a clever setup by only offering 20 to 1 odds. What do you think the actual odds are of completing the whole deck successfully?

Recently, I took a trip to Las Vegas and found a unique blackjack game known as the 'World’s Most Liberal Blackjack' at the Las Vegas Club. This game allows players to double down with any two, three, or four card combinations, split and re-split aces as many times as desired, and split and re-split any pair indefinitely. If you surrender your first two cards, you only lose half of your bet, and a hand with six cards wins automatically. However, there's a rule that blackjack only pays even money unless it’s suited, in which case it pays out 2 to 1. Should I consider this game more favorable than a traditional 3 to 2 blackjack using six decks with the dealer standing on a soft 17? Additionally, would doubling down still be advantageous if the payout for blackjack is only even money?

The house edge for this game is approximately 1.30% or 1.33%, depending on whether the game uses five or eight decks. It’s clear that any game where blackjack pays 3 to 2 offers better odds. If you were to play this particular game—which I would personally advise against—it’s vital to always stand on blackjack. In my opinion, the claim that this is the 'World’s Most Liberal Blackjack' is misleading advertising.

Suppose you found a casino offering the option for a player to bet on both the player and banker at the same time in baccarat. Would there be any strategic advantage to doing this? Also, what if they took your total bet into consideration when rating you (for instance, placing $25 on the Banker and $25 on the Player, so you're rated for $50 in total)?

Several casinos allow players to place behind bets on the blackjack table, creating a unique dynamic. Could you explain the optimal strategy for splitting pairs when the behind wager greatly surpasses the main wager, especially considering that both bettors are collaborating?

How many sevens would need to appear in 500 rolls for one to assert with less than a 2.5% chance that the outcome was purely random and not statistically significant?

I appreciate your encouraging feedback. It’s crucial to express that the probability of the outcomes not being random should be framed as the likelihood that a random game would lead to such results. No one anticipated that the 500 rolls would definitively prove anything one way or the other. Although I didn’t set the line at 79.5 sevens, I doubt it was established based on statistical significance; it might have just been a point that both sides accepted for the wager. Stanford Wong Experiment The 2.5% significance level equates to 1.96 standard deviations from the mean. You can calculate this using the formula =normsinv(0.025) in Excel. The standard deviation for 500 rolls is determined as sqrt(500*(1/6)*(5/6)), yielding a result of 8.333. Thus, 1.96 standard deviations below expectations would be 1.96 * 8.333 = 16.333 rolls under the expected value. The anticipated count of sevens within 500 rolls is 500*(1/6), which equals 83.333. Hence, 1.96 standard deviations below that yield a threshold of 83.333 - 16.333 = 67. When using the binomial distribution, the precise chance of rolling 67 or fewer sevens stands at 2.627%.

What is the average number of rolls required to achieve a Yahtzee?

Assuming the player consistently holds onto the number that appears the most, the average is around 11.09 rolls. A table is available showcasing the distribution of the number of rolls over a random simulation involving 82.6 million trials.

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