In the Station casinos, there is the "Big 3" game in their bingo rooms. You must get 3 of the first 4 numbers to win a progressive jackpot. What are the odds of that? Thank you.

Joseph G. from Las Vegas

For the benefit of other readers, the Big 3 is a bingo side bet at all the Station Casinos and the Fiesta Rancho. The player is given a ticket, either paper or loaded into an electronic unit, with three random bingo numbers out of the 75 possible. If the first four bingo numbers called in that session contain all three of the player’s numbers, then the player will win a progressive jackpot. The jackpot starts at \$1000 at grows by \$200 a day until somebody wins. Every session, and property, has an independent jackpot.

The number of winning combinations is 72, because three of the balls must match, and the fourth can be any one of the other 72 balls. There are combin(75,4) = 1,215,450 possible combinations. Thus, the probability of winning is 72/1,215,450 = 0.000059. The player can buy 48 tickets for \$10, thus the cost per ticket is 10/48 = 0.208333 dollars. The breakeven meter, where there is zero house edge, is (10/48)/(72/1,215,450) = \$3,516.93.

Station Casinos indicate the Big 3 Jackpots on their . There you will see the meter often will exceed \$3517. When I answered this question on August 30, 2007, two of the eight properties had a player advantage, the Palace Station and Fiesta Rancho. This is one of the few bets in Las Vegas that often have a player advantage. Unfortunately, they limit the number of cards you can buy, making it not worth the bother to most people, including me, to make a special trip.

I know that the rules of Spanish 21 say that the "envy bonus" is always \$50, and the super bonus is \$1,000, for bets of \$5 to \$25, or \$5000, for bets of \$25 or higher. I was curious, what is the house edge penalty for playing at an empty table, or betting more than \$25 a hand? Casinos offer regular blackjack for high rollers without maximum payouts. Do you know of any casinos that offer good rules for high rollers in Spanish 21? (i.e. a \$500 envy bonus for a bet of \$50)

Dean from Toronto

I don’t like it when games give worse odds to the higher bettors either. The value of these Super Bonuses is almost zero. The probability of hitting the Super Bonus is one in 549,000 with eight decks, and one in 668,000 million with six decks. Assuming six decks, the value of the envy bonus is worth 0.0015% per additional player, besides yourself. Sorry, I don’t know of any casinos that sweeten the bonuses for larger bets.

I played 50-line 9/6 Jacks or Better \$1 machine over the weekend and got killed. Any idea what the odds are of putting \$800,000 coin in on 50-line \$1 and not hitting a single royal? Just curious.

Dave from Las Vegas

If you were playing single line it would be easy. \$800,000 is 160,000 \$5 hands. That is 3.9616 royal cycles. The probability of no royals can be closely approximated as e-3.9616 = 1.9%.

The math gets messier with mutli-line games. I think the easiest way to answer the question is by random simulation. My video poker appendix 6 shows the probability of getting at least one royal per hand in 50-play 9/6 Jacks or Better is 0.00099893. Each hand of \$1 50-play costs \$250. So you would have played 3,200 initial hands. The expected number of hands with a royal in 3,200 hands is 3.1966. By the same method of approximation, the probability of getting zero royals is e-3.1966 = 4.09%. The exact answer, based on the simulation results, is (1-0.00099893)^3200 = 0.04083732, or 4.08%.

With regards to your California Three Card Poker section, there is a casino in Oceanside that offers the game with similar rules. The casino shows one card, but in order to win your ante bet the dealer hand does not have to beat the player’s hand. What is the house edge in this game?

Joshua K. from Oceanside

That rule change is worth 2.49% in the player’s favor, lowering the house edge from 4.30% to 1.80%. The subraction is not exactly 2.5% less, due to rounding.

Is it appropriate to tip a host? Recently, while staying at an off-Strip hotel, my wife and I were approached by the host who comped our hotel room for four nights. We were invited to return to that hotel for another complimentary stay. Should we tip the host? What would be an appropriate amount?

Jack N. from Eastpoint, MI

The etiquette on this is not set in stone, so the following is just my opinion. Tipping hosts is, most of the time, truly voluntary and not expected. If you do tip, it should not be in cash. Gift certificates, sport book tickets, or physical items are acceptable. Some people believe that hosts will work a little harder for you if you tip. Personally, I haven’t noticed a difference. At times when I gave a host an envelope with a gift certificate in it, the host seemed uncomfortable accepting it, but other times not. The best way to make your host happy is to play hard in the casino. Hosts are judged according to how much their players play vs. how much they give out. It doesn’t make them look good if you squeeze them for everything you can get, and then don’t play commensurately in the casino. An exception to the general rule about tips being not expected is that if a host gets you into a tournament, and you win a lot of money, then you should tip both the dealers and your host generously.

If I make 1,000,000 spins on the event that has 1 in 1,000,000 chance of winning, what are my chances of winning at least once?

Ares75 from Petrovce

If the probability of winning is 1/n, and you play n times, as n approaches infinity, the probability of winning at least once approaches 1-(1/e), where e = 2.7182818..., or about 63.21%. The exact answer can be expressed as 1-(999,999/1,000,000)1,000,000 = 0.63212074. My estimate is 1-(1/e) = 0.63212056, which agrees to six decimal places.