Ask The Wizard #293

Yes! In joker poker Typically, when dealing with a weak hand, it's advisable to hold onto one of the middle cards that has the highest likelihood of completing a straight or a flush during the draw. Nevertheless, there are instances when the decision is quite close or even evenly matched regarding which card to keep.

For instance, in the 97.19% payout structure of joker poker with two pairs and a hand showing Qh, 10d, 5d, 7c, and 2c, the best moves are tied between retaining just the 10d, 5d, or 7c, yielding an expected value of 0.240703 for all three scenarios. This can be confirmed with my analysis. video poker hand analyzer .

Here are additional examples of similar strategic plays for the same joker poker payout structure:

• QC 10S 5D 7H 2C
• QH 10D 5D 7H 2C
• KH 10D 5D 8C 3C
• KC 10S 5D 8H 3C
• KH 10D 5D 8H 3C

I appreciate Gary Koehler for his assistance with this query.

For the benefit of my readers, I’ve developed a strategy that capitalizes on the extra half points from point spreads of 3 and 7 when using half-point parlay cards. I’ve demonstrated that this method can yield reliable, albeit highly variable, advantages.

The recent rule adjustment pushes the extra point attempt back to the 15-yard line. This change is likely to decrease the success rate of the kick and encourage more attempts for two-point conversions. With an increase in two-point conversion attempts, regardless of whether they succeed, we may see fewer games concluded by margins of three or seven points, which in turn diminishes the importance of those extra half points in half-point parlays. Should we be concerned? Let's explore.

Initially, we should have some concerns, but this specific rule change might not be the primary cause. Other rule modifications have already inhibited defensive play, resulting in higher scoring matches. The graph below illustrates the average points scored per game for each season from 1994 to 2014.



As the graph indicates, the average score per game remained around 42 up to 2006. However, from 2007 onward, this average has gradually increased by approximately half a point each year. This prompts the question of whether the rising average score is flattening the distribution of winning margins, particularly concerning the crucial numbers of 3 and 7. The next graph displays the probabilities of victory margins of three and seven points across the years.



From the visual evidence, it is clear that the likelihood of achieving a seven-point victory margin has remained stable at 9.1%. Conversely, the probability of a three-point margin has been consistently decreasing, notably since 2004. This trend is troubling, given the significant number of point spreads of three within the NFL.

Regardless of the factors contributing to the decline in three-point victory margins, the appeal of half-point parlay cards appears to be waning. Will the new regulation regarding extra points exacerbate this issue?

In my view, this change might have a detrimental effect, but the impact should be minimal. Prior to this adjustment, my analysis suggested that approximately 4.8% of touchdowns were followed by attempts at two-point conversions. As I reflect on this, the 2015 season has only seen two weeks of gameplay after the regulation change, with 167 touchdowns scored and 15 of those followed by attempts for two-point conversions, resulting in an attempt rate of 9.0%.

From a mathematical perspective, I believe pursuing a two-point conversion should become a more frequent strategy, particularly for teams that often seek variance in outcomes. I find it perplexing why this isn't the case. Although it deviates from your inquiry, I felt compelled to express this thought.

That's an interesting query. The advantage in straight betting at odds of 11 to 10 is 3.09%, while the parlay's advantage stands at 10.22%. This would suggest that the parlay may be more favorable.

However, it is important to remember that the most effective strategy for bankroll growth when employing any type of advantageous play is to utilize the Kelly Criterion. Kelly Criterion According to the Kelly strategy, the ideal bet size should maximize the expected logarithm of the bankroll after the wager is made. For bets with only two possible outcomes, the optimal bet size conveniently turns out to be the advantage divided by the payout in a 'to one' format.

Straight bets yield payouts of 10 to 11. Consequently, the optimal bet size would be calculated as 0.030909 divided by (10/11), which equals 0.034000. With a player advantage of 3.09%, you can expect to gain approximately 0.001051 times your bankroll for every game played through straight betting.

For parlay bets, the optimal bet size is 0.102248 divided by 6, resulting in 0.017041. With a player advantage of 0.102248, the expected gain would be around 0.001742 times the bankroll from straight betting.

Nevertheless, straight betting remains the superior option since it allows for three opportunities to wager. To compute the anticipated return from flat betting, you can multiply the expected gain of each bet by three, resulting in 3 × 0.001051 = 0.003153. This amount is 81% greater than the growth in bankroll expected from parlay betting.

If your bankroll has reached a level where you hit the maximum betting limits, it would be wiser to place a parlay bet first, as they generally do not affect the betting lines, and later proceed with your straight bets.

This topic is explored and discussed in my forum at Wizard of Vegas .

The situation presents a challenging decision, which I believe is precisely what the NFL intended to achieve with this new rule. Several factors should be weighed when making this choice:

• Probability of success by kicking.
• The likelihood of success when opting for a two-point conversion.
• The overall probability of victory for each team.

Toward the end of the game, it becomes crucial to consider the significant scoring margins. For example, if time is running out and your team has just scored a touchdown to take a three-point lead, kicking the extra point may be the best move. If successful, this would put you up by four points, necessitating the other team to score a touchdown to win. However, if you choose to attempt a two-point conversion and fail, the other team could potentially equalize with just a field goal. Therefore, I will restrict my analysis to the earlier stages of the game when crucial scoring margins are less impactful.

To provide an answer to your question, I created a simulation program. Although it's quite basic, I believe it accurately reflects the dynamics of the NFL concerning field goals, touchdowns, and turnovers.

According to the article What Consequences Can We Expect from Relocating Extra Points to the 15-Yard Line? According to Kevin Rudy, the probability of successfully converting the extra point from the 15-yard line is 94.2%. For simplicity in my calculations, I assumed a success rate of 94%.

There is more debate regarding the success rates of two-point conversions, which could change significantly based on the specific offense and defense involved. To account for this variability, I conducted simulations with success probabilities ranging from 46% to 50%, testing in 1% increments.

In each simulation, one team consistently kicked after a touchdown, while the opponent attempted two-point conversions at varying success probabilities. Additionally, I ran scenarios where both teams opted to kick, allowing for a comparison against two-point conversion outcomes.

Table 1 below illustrates the results between two similarly skilled teams. Team A always opts for a kick after a touchdown. This table presents the likelihood of Team B winning and the average additional points scored per game based on whether Team B chooses to kick or attempt a two-point conversion (2PC), along with the associated probability of success.



It is notable that at a success probability of 47%, the expected extra points remain at 0.94, regardless of whether Team B kicks or attempts a two-point conversion. However, the probability of winning diminishes by approximately 0.1% if opting for the two-point conversion. My assumption is that if Team B misses their conversion, Team A could have a chance to tie with two field goals. It should be noted that this effect related to significant scores is relatively small but still tends to favor a conservative strategy of kicking, all else being equal. The bottom line indicates that if both teams are evenly matched, the success probability of a two-point conversion should ideally be at least 49% in order to warrant an attempt.

Table 2 below displays the outcomes for two teams with differing strengths, where Team A is the more capable contender. Team A exclusively kicks after a touchdown. It provides the same statistics as presented in Table 1.

According to Table 2, the weaker team should pursue the two-point conversion at all the listed success probabilities. This remains true even at a 45% success rate, where the anticipated total points scored is lower. You may wonder why this is the case. It is because the underdog team should aim to increase variance, whereas the stronger team should strive to minimize it.

Table 3 below reports the outcomes for two teams where Team B is stronger, with Team A again always kicking after scoring a touchdown. The data corresponds to those found in Tables 1 and 2.

According to Table 3, the probability of Team B winning by choosing to kick stands at 71.17%. This probability remains lower than that of all success scenarios outlined, despite the expectation of higher total points when achieving a 49% or 50% success rate. The underlying rationale is that weaker teams should pursue variance while stronger teams should steer clear of it. The objective of the two-point conversion attempt introduces unpredictability to the game, making it more appealing for weaker teams to attempt them, all factors being equal, as opposed to stronger teams.