Ask The Wizard #329

By comparison, the chances of hitting a royal flush in standard video poker typically range from about 1 in 40,000 to 45,000, under ideal conditions. Here are some specific statistics for a few randomly selected games:

• 9-6 Jacks or better: 1 in 40,391
• 25-15-9-4-4-3 (Illinois) Deuces Wild — 1 in 43,423
• 9-7 Triple Double Bonus: 1 in 45,358

For players chasing a royal flush, the odds improve significantly in Chase the Royal . This is an earlier version of video poker where users can swap a pair of face cards dealt for three cards that lead to a royal flush. To make the exchange worthwhile, the game enhances the payout for straights and flushes if you opt to trade. The precise odds for a royal flush vary based on the foundational game and its payout structure. The chances peak with a foundational game of Triple Bonus and an 8-5 payout structure, achieving a royal frequency of 1 in 9,151. This figure encompasses both royal flushes achieved by drawing, which pays 800 to 1 with a frequency of 1 in 9,282, and those dealt initially, paying 2000 to 1 at a frequency of 1 in 649,773.

However, the situation improves further when examining games that require players to pay a fee equating to their initial bet to activate a special feature. In Draw Poker with Dream Card (not to be confused with Dream Card Poker ), players often receive their desired card (assuming a mathematician is envisioning it) as the fifth card in their hand. The probability of achieving the Dream Card peaks in Jacks or Better, reaching 50.5%. In the case of 9-6 Jacks or Better, the overall frequency for a royal flush stands at 1 in 8,105. It's important to remember that with the fee applied to activate the feature, the effective payout for a royal flush drops to 400 to 1.

Speaking of Dream Card Poker (distinct from Draw Poker with Dream Card), the frequency of hitting a royal flush in that game isn’t as favorable. The highest frequency appears to be in 11-8-6 Jacks or Better with odds of 1 in 15,034.

This answer does not consider Movin' On Up Poker , an old-school and lesser-known video poker game allows players two or three draws instead of just one. The exact probability for a royal in that version is unclear, but in Triple Draw, where players must pay a fee equal to five times their initial wager for the extra draws, I estimate it to be around 1 in 4,000.

To sum it up, excluding games where players must pay an additional fee for unique features, my recommendation is to pursue Chase the Royal.

In my research, I discovered that this isn’t actually a side bet, but rather what the pick-20 ticket awards for catching no numbers. Below is the comprehensive breakdown of my analysis regarding the pick-20 ticket at Station Casinos.

The bottom right cell indicates that the overall return for this ticket is 69.70%, which is standard for live keno.

To address the question regarding catching zero, the probability column indicates that this is at 0.001186, and with a payout of 200 for 1, it contributes 23.71% towards the overall return.

Good question. Here are my answers:

Here is my solution to both (PDF).

That is indeed the case. The formula has been evolving to account for the rising full retirement age, which has shifted from age 65 to 67, alongside incentives for delaying retirement (delayed retirement credits). To keep things straightforward, the following explanation will focus only on individuals born in 1960 or later, by which time the full retirement age is set at 67.

Before proceeding, let’s define the monthly benefit that one would receive at the full retirement age, referred to as the Primary Insurance Amount (PIA). The earliest age to begin claiming retirement benefits is 62, while the latest is 70. You can delay benefits beyond age 70, but doing so no longer provides additional financial incentive.

If you apply between 1 and 36 months before reaching full retirement age, the monthly benefit will be reduced by 1/180 for each early month. For instance, if you apply precisely at age 65, which is 24 months early, your monthly benefit would be lowered by 24/180, amounting to a reduction of 13.33%.

For applications made 37 to 60 months early, the monthly benefit drops by 36/180, or 20%, during the first 36 months prior to reaching 67, plus an additional 1/240 for each month before turning 64. For example, if you apply exactly at age 62, or 60 months early, the monthly benefit would get cut by (36/180) + (24/240), equating to a total reduction of 30%.

What if one applies after the age of 67? If you claim benefits after that age, your monthly benefit rises by 1/150 for each additional month. For instance, if you wait the full 36 months until age 70, your monthly benefit will increase by 36/150, resulting in a 24% rise.

What does the mathematics suggest about when to apply for benefits to maximize overall payments? Fortunately, because of cost-of-living adjustments, we can disregard inflation for simplicity. Additionally, let’s not consider potential eligibility for other benefits, such as spousal and widow benefits, as these would complicate matters significantly. Ignore the Earnings Test, which would penalize benefits for income surpassing certain limits. Therefore, in a straightforward one-person context, the optimal timing really hinges on your expected lifespan. The following table illustrates the ratio of monthly benefit to Primary Insurance Amount based on the age benefits commence.

It is evident that if you anticipate living to an advanced age, it’s advantageous to delay your application, whereas if you expect to pass away shortly after turning 62, it would be wise to take benefits as soon as you can. The question is, where is the tipping point? The next table will display the optimal age for applying in order to maximize total benefits according to your projected age of death.

In simple terms, if you're expecting to live until age 76 or younger, it is advisable to apply for benefits as early as possible. Conversely, if you believe you will live past age 86, waiting until you’re 70 to apply would be best. For those in the intermediate range, where most individuals typically pass away, refer to the table. For example, statistics indicate that the average male at 62 expects to live until approximately 81.19, while the average female is expected to live until about 84.13. This means the average male should ideally apply at around 66 and a half years of age, and the average female at approximately 69 and a quarter. actuarial calculator Certainly, everyone’s health and financial circumstances differ, so it's important to take all these factors into account when deciding when to claim retirement benefits.

Strategies and insights grounded in mathematics for various casino games like blackjack, craps, roulette, and many others.