Ask The Wizard #280

Please see my companion site MathProblems.info, problem number 210, for the answer and solution.

Not counting pushes, the probability of getting at least 88 wins out of 139 picks is 0.00107355, or 1 in 931. This is pretty underwhelming. I'm sure there are 930 other animals out there who did worse that nobody writes about. For more information on Princess, read the article N.J. camel predicts Giants over Patriots at ESPN.com.

Let's start by looking at some no-nonsense video poker games. The following table shows the probability of a royal, from highest to lowest. This table does not count wild royals, which pay much less than a natural.

Surprisingly, Joker Poker is the most likely to get a natural royal. This surprises me because of the extra card, which can't be used to make a natural royal.

Next, let's look at some non-standard video poker games.

In 9-6 Jacks Royal Draw, the odds are 1 in 12,178.

In 7-5 Jacks Second Chance Royal, the odds are 1 in 10,827. This includes "second chance" royals, which pay 200 only.

Finally, I think an honorable mention should be made for Triple Double Bonus, where four aces plus a 2-4 kicker pay the same 800 per coin bet as a royal flush. The odds of either 800 win are 1 in 10,823, based on the 9-7 pay table. Even better is Royal Aces Bonus Poker, which pays 800 for any royal flush or four aces, for a probability of 1 in 3,673 for a win of 800.

However, all things considered, my answer for the game with the most natural royals is Chase the Royal. Based on the 9-6 Jacks pay table, the probability of a royal flush is 1 in 9,084.

Yes. Remember these numbers: 8, 27, 47, 67. Here is what they mean.

• If the Banker's total is 3, and the Player draws anything except an 8, then Banker draws.
• If Banker's total is 4, then the Banker draws against a Player third card of 2 to 7.
• If Banker's total is 5, then the Banker draws against a Player third card of 4 to 7.
• If Banker's total is 6, then the Banker draws against a Player third card of 6 to 7.

I hope you're happy; I spent hours on this.

To answer the question it is important to quantify behavior under the Chelsea Handler red head hypothesis. Here are my assumptions.

1. A red head will never mate with another red head.
2. The female will always choose the male to mate with.
3. Everybody will mate and each mating will produce the same number of children.
4. The female redheads will get first dibs at a mate, choosing randomly among the non-red heads.
5. The female carriers (with one red-haired gene) will choose a mate randomly among the men left over by the red heads.
6. The negative females (neither red-haired gene) will chose randomly among the men left over by the red heads and the carriers.

I start with a red-haired probability of 4%, according to Today I Found It. I then assume that prior to now there was no bias against red heads.

Assuming the bias against red heads starts with the next generation, and continues, what will be the trend of the total population to have red hair? After a lot of work in a spreadsheet, which I won't get into, here are the first eight generations, starting from this one.

What we see is that by the third generation the proportion of the population with red hair will converge to 3.90%. So, despite what Chelsea may say, I think the red heads have nothing to worry about.

This question was raised and discussed in my forum at Wizard of Vegas.