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Ask the Wizard #329

If I am using perfect betting strategy for video poker play, which game gives me the best chance of getting a royal flush?

"Anonymous" .

As a basis of comparison, the probability of a royal flush in conventional video poker ranges from about 1 in 40,000 to 45,000, assuming optimal strategy. Here are some exact figures for some 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 Royal hungry players, the probability goes up significantly in Chase the Royal. This is an early video poker variant where the player may exchange a dealt pair of face cards for three to a royal flush on the deal. To make it a good value to trade, the game bumps up the win on the straight and flush, if you switch. The exact royal probability will depend on the base game and pay table. The probability is maximized with a base game of Triple Bonus and 8-5 pay table, at a royal frequency of 1 in 9,151. This includes both royals on the draw, which pay 800 for 1, at a frequency of 1 in 9,282 and on the deal, which pay 2000 for 1, at a frequency of 1 in 649,773.

However, it gets even better if you consider games in which the player must pay a fee equal to his base wager to enable a feature. In Draw Poker with Dream Card (not to be confused with Dream Card Poker), the player often gets the card of his dreams (assuming its a mathematician doing the dreaming) as the fifth card on the deal. The Dream Card probability of maximized in Jacks or Better, at 50.5%. In 9-6 Jacks or Better, the overall Royal frequency is 1 in 8,105. Keep in mind that with the fee to enable the feature, the effective win on a Royal drops to 400 for 1.

As long as I mention Dream Card Poker (a different game than Draw Poker with Dream Card), the Royal frequency in that game is not as high. It appears to be highest in 11-8-6 Jacks or Better at 1 in 15,034.

This answer does not consider Movin' On Up Poker, which is an old and obscure video poker game where the player gets two or three draws, instead of one. I don't know the Royal frequency in that game, but in Triple Draw, in which the player must pay a fee equal to five times his base wager to enable the extra two draws, I roughly estimate it to be about 1 in 4,000.

In conclusion, if you don't count games where the player must pay an extra fee to enable some kind of gimmick, my answer is Chase the Royal.

At the Santa Fe Station, there is a side bet in pick-20 keno that pays 200 for 1 for zero catches. What are the odds?

Michael

Upon doing some research, I found this isn't a side bet, but what the pick-20 ticket pays for catching zero. The following is my complete analysis of the Station Casinos pick 20 ticket.

Station Casinos Pick 20 Keno

Catch Pays Combinations Probability Return
20 50000 1 0.000000 0.000000
19 50000 1,200 0.000000 0.000000
18 50000 336,300 0.000000 0.000000
17 50000 39,010,800 0.000000 0.000001
16 10000 2,362,591,575 0.000000 0.000007
15 8000 84,675,282,048 0.000000 0.000192
14 4000 1,940,475,213,600 0.000001 0.002196
13 1000 29,938,760,438,400 0.000008 0.008468
12 200 322,309,467,844,650 0.000091 0.018234
11 20 2,482,976,641,173,600 0.000702 0.014047
10 10 13,929,498,956,983,900 0.003940 0.039401
9 5 57,559,913,045,388,000 0.016281 0.081407
8 2 176,277,233,701,501,000 0.049862 0.099724
7 1 400,535,252,907,552,000 0.113295 0.113295
6 0 672,327,031,666,248,000 0.190175 0.000000
5 0 824,721,158,843,931,000 0.233281 0.000000
4 0 724,852,581,015,174,000 0.205032 0.000000
3 0 441,432,713,697,822,000 0.124864 0.000000
2 1 175,755,617,490,799,000 0.049714 0.049714
1 2 40,896,043,959,078,000 0.011568 0.023136
0 200 4,191,844,505,805,500 0.001186 0.237141
Total 3,535,316,142,212,170,000 1.000000 0.686961

The lower right cell shows the overall return of the ticket is 69.70%, which is typical for live keno.

To answer the question about catching 0, the probability column shows the probability of that is 0.001186 and at a win of 200 for 1, it returns 23.71% towards the return.

What is the average area and perimeter of a rectangle randomly inscribed in a circle of radius 1?

USpapergames

Good question. Here are my answers:

The average perimeter is 16/π = apx 5.0929581789406507446042804279.

The average area is 4/π = apx 1.27323954473516268615107010698.

Here is my solution to both (PDF).

I understand that Social Security will cut your retirement benefits if you retire early and increase them if you retire late. That said, when should I retire to maximize the total amount I receive?

"Anonymous" .

That is indeed true. The formula has been changing to adjust for the increasing full retirement age from age 65 to 67 and incentives (delayed retirement credits) to applying late. To keep things simple, this answer will apply only to those born in 1960 or later, when the full retirement age reaches 67.

Before going on, the monthly benefit at exactly the full retirement age is called the Primary Insurance Amount, or PIA. The earliest one may apply for retirement is age 62 and the latest is age 70. One can wait past age 70, but there is no longer a reward for doing so.

If applying 1 to 36 months early, the montly benefit will be cut by 1/180 per month. For example, if applying exactly at age 65, 24 months early, the montly benefit would be cut by 24/180 = 13.33%.

If applying 37 to 60 months early, the montly benefit will be cut by 36/180, or 20%, for the first 36 months before age 67, plus 1/240 for each month before age 64. For example, if applying at exactly age 62, 60 months early, the monthly benefit would be cut by (36/180) + (24/240) = 20% + 10% = 30%.

What about applying late? If applying after age 67, the monthly benefit will be increased by 1/150 per month. For example, if applying the full 36 months late, at age 70, the montly benefit will be increased by 36/150 = 24%.

What does the math say about when to apply to maximize total benefits? Fortunatley, with cost of living adjustments, we can ignore inflation, to keep things simple. Let's also ignore potential entitlement to other benefits like spousal and widows, which make things really complicated. Let's also ignore the Earnings Test, which will cut benefits for work income above certain threshholds. That said, in the simple one-person situation, the answer depends on when you expect to die. The following table shows the ratio of the montly benefit to the Primary Insurance Amount, according to the exact age benefits began.

Benefit Ratio

Age (Years) Age (Months) Benefit Ratio
62 0 70.00%
62 1 70.42%
62 2 70.83%
62 3 71.25%
62 4 71.67%
62 5 72.08%
62 6 72.50%
62 7 72.92%
62 8 73.33%
62 9 73.75%
62 10 74.17%
62 11 74.58%
63 0 75.00%
63 1 75.42%
63 2 75.83%
63 3 76.25%
63 4 76.67%
63 5 77.08%
63 6 77.50%
63 7 77.92%
63 8 78.33%
63 9 78.75%
63 10 79.17%
63 11 79.58%
64 0 80.00%
64 1 80.56%
64 2 81.11%
64 3 81.67%
64 4 82.22%
64 5 82.78%
64 6 83.33%
64 7 83.89%
64 8 84.44%
64 9 85.00%
64 10 85.56%
64 11 86.11%
65 0 86.67%
65 1 87.22%
65 2 87.78%
65 3 88.33%
65 4 88.89%
65 5 89.44%
65 6 90.00%
65 7 90.56%
65 8 91.11%
65 9 91.67%
65 10 92.22%
65 11 92.78%
66 0 93.33%
66 1 93.89%
66 2 94.44%
66 3 95.00%
66 4 95.56%
66 5 96.11%
66 6 96.67%
66 7 97.22%
66 8 97.78%
66 9 98.33%
66 10 98.89%
66 11 99.44%
67 0 100.00%
67 1 100.67%
67 2 101.33%
67 3 102.00%
67 4 102.67%
67 5 103.33%
67 6 104.00%
67 7 104.67%
67 8 105.33%
67 9 106.00%
67 10 106.67%
67 11 107.33%
68 0 108.00%
68 1 108.67%
68 2 109.33%
68 3 110.00%
68 4 110.67%
68 5 111.33%
68 6 112.00%
68 7 112.67%
68 8 113.33%
68 9 114.00%
68 10 114.67%
68 11 115.33%
69 0 116.00%
69 1 116.67%
69 2 117.33%
69 3 118.00%
69 4 118.67%
69 5 119.33%
69 6 120.00%
69 7 120.67%
69 8 121.33%
69 9 122.00%
69 10 122.67%
69 11 123.33%
70 0 124.00%

Clearly, if you plan to live to a ripe old age, it pays to apply late and if you expect to not make it much past 62, then grab what you can while you can. The question is, where is the indifference point? The following table shows the optimal age to apply to maximize total benefits according to your age at death.

Optimal Age to Apply

Age at Death Age to Apply (Years) Age to Apply (Months)
76 62 0
77 62 6
78 63 0
79 63 6
80 66 0
81 66 6
82 68 3
83 68 9
84 69 3
85 69 9
86 70 0

In simple English, if you expect to die at age 76 or before, apply as early as possible. If you expect to die at age 86 or later, wait until you're 70 to apply. In the middle range, when most people actually do die, consult the table. As an example, according to my actuarial calculator, the average man at 62 can expect to die at age 81.19, while the average woman will die at age 84.13. Thus, the average man should apply at an age of 66 and 6 months and the average woman at 69 and 3 months.

Of course, everybody's health and financial situation is different, so carefully consider everything in deciding when to apply for retirement.