Ask The Wizard #315

Before diving into the analysis, I must clarify that the insights shared are derived from statistical averages. Each contestant should consider their familiarity with the Final Jeopardy category and gauge the likelihood of their opponent answering correctly.

To provide a thorough response, I analyzed data spanning four seasons from the show. Jeopardy Archive This investigation examined the four potential scenarios involving the first-place participant and the second-place contestant regarding their performance in Final Jeopardy, both correct and incorrect.

Before proceeding, it's essential to establish some definitions:

Let x represent the likelihood that the leading player opts for a high bet.
Let y indicate the likelihood that the player who is chasing chooses a high bet.
The function f(x,y) will represent the chance of the player who bets high winning the game.

Next, that's how we can express f(x,y) using the x and y values derived from the previously mentioned table:

f(x,y) = 0.823xy + 0.545x(1-y) + 0.468(1-x)y + (1-x)(1-y)
f(x,y) = 0.810 xy - 0.455x - 0.532y + 1

To discover the most advantageous values for x and y, we should compute the derivative of f(x,y) in relation to both x and y.

f(x,y) d/dx = -0.455 + 0.810y = 0
Thus y = 0.455/0.810 = 0.562

f(x,y) d/dy = -0.532 + 0.810x = 0
Thus x = 0.523/0.810 = 0.657

Consequently, the player in the lead should bet high with a probability of 65.7%, while the player trailing should also opt to bet high with a probability of 56.2%.

From my observations, it seems that the leading player actually bets high more than 65.7% of the time, so if I were in the second position, I would choose to bet low.

When both players follow this randomized approach, the chances of the leading player winning stand at 70.1%.

Setting aside theoretical considerations, if you're in the lead, try to anticipate what the chasing player will do and mirror that. If you're the one chasing, predict the leading player’s move and then do the opposite. This is a universal strategy applicable to all such tournaments.

This topic has been opened for discussion on my forum at Wizard of Vegas .

For the benefit of other participants, here’s the line for Grand Slams for May 11, 2019 . The headers in the columns should be labeled 'yes' and 'no', rather than 'home' and 'away'.

To respond to this query, I scrutinized every MLB matchup from the 2011 through 2018 seasons, during which there were 858 Grand Slams recorded across 19,440 games. This results in an average of 0.0441 Grand Slams per game. It’s important to note that this average doesn't equate to the probability of any single game having a Grand Slam, as one game may include multiple instances. Applying the Poisson distribution, the chance of any game having at least one Grand Slam is 4.3176%.

Given this probability, it's straightforward to formulate a table that displays the likelihood and fair line for experiencing at least one Grand Slam based on the number of games played.

I must emphasize that my conclusions are based on averages from MLB. On any specific day, players should take into account the individual games occurring. Notably, games with a higher over/under line correlate to a greater likelihood of witnessing a Grand Slam.

Given that advantage and the total number of hands played, the projected win would amount to 2,015 units. Assuming a standard deviation of 1.1 per hand, the overall standard deviation for our play results in 1,561. The difference between your actual results and the expected win comes to 3,490. Calculating this gives you 3,490 divided by 1,561, yielding 2.24 standard deviations. The probability of obtaining results this poor or worse stands at 1.27%.

The chance of achieving a royal flush is quantified as 1 in 32,487. Each occurrence results in shortchanging you by $20,000, which equates to 400 Ante bets. This translates into a cost of 400 divided by 32,487, or approximately 1.23% of all wagers made on the Ante. As a result, this increases the casino's edge from 2.185% to 3.416%, when calculated against the Ante bet.

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