Ask the Wizard #233

There are 6³=216 ways to roll three dice. Six of those combinations will result in a three of a kind (1-1-1 to 6-6-6). So the probability of a three of a kind is 6/216 = 1/36. There are four possible spans for a straight (1-2-3 to 4-5-6). There are also 3!=6 ways to arrange the three dice in a straight. So, there are 4*6=24 straight combinations. Thus the probability of a straight is 24/216 = 1/9.

According to Life: the Odds (and How to Improve Them) by Gregory Baer, the odds of a hole in one on a par 3 hole in the PGA tour is 1 in 2491. I believe those distances fall in the par 3 range.

A 1 handicap is darn good, so I'm not going to give much of a discount compared to PGA Tour players. Let's say your son's probability per par 3 hole is 1 in 3,000. A typical gold course will have about four par 3 holes. Let’s say your son plays every day. That would be 28 par 3 holes a week. The probability of making exactly two hole in ones would be combin(28,2)×(1/3000)²×(2999/3000)²⁶ = 1 in 24,017.

Not that you asked, but you have a 43.4% advantage if your first card is a jack. It is the dealer’s fault for flashing the card. Contrary to what some members of the casino staff, especially in security, incorrectly believe, you are legally allowed to make use of whatever information made available to you under normal playing conditions.

Morally, you should follow your own conscience. You have to live your own life. That said, I think most players, including me, would be okay with increasing the bet in that situation. For one thing, game security is not the player’s job. For another, the casinos take advantage of, if not rely on, player mistakes. For example, consider the big 6/8 bet in craps. The casinos have no compunction about accepting a bet on that, when the place bet on 6 or 8 pays on exactly the same thing, but has better odds. See if you are offered forgiveness if you foul your hand in pai gow poker, even if the correct setting is totally obvious.

If it happens again, don’t get too greedy, and act nonchalant. If you suddenly go from a $10 to a $500 bet, it will set off all kinds of red flags. A good dealer would realize why, and ultimately the bet would not be accepted, or a card would be burned.

Nice find! You didn’t say what denomination you are playing, which is important, so I’m going to assume dollars. For five-coin maximum bet, the number of doubles required for a win of w (where w<1200) is 1+int(log(1200)-log(w))/log(2).

The following table shows for each initial hand the pre-double win, pre-double probability, number of doubles required, post-double win, and probability achieving the post-double win, including the $250 bonus. The lower right cell shows a return of 115.5%. You will get a jackpot every 297 hands on average, with an average jackpot of $1,717.46.