A shuffle through the gaming mailbag

Q. This isn't really about a casino game, but I have a question about odds. My brother-in-law and I were cutting cards to pass time before the start of a game we were going to watch on TV. High card wins, and the first to win 10 times is the overall winner. Ties didn't count.

We both had a couple of streaks. I won four in a row at one point. Near the end, I'd won nine times and he'd won eight. Then he won the last two and beat me. I was wondering what the odds of my winning were when I had the 9-8 lead. How big an upset was it? Also, what were the odds of my winning four times in a row?

A. Since you weren't counting ties, the odds in your game were just the same as flipping a coin. On every decision, you have a 1 in 2 chance of winning. It's an even game.

When you led 9-8, the maximum number of decisions left that would get one of you to 10 wins was two. In any two-decision set, there are four possible outcomes. Looking at it from your perspective, they're win-win, win-loss, loss-win, loss-loss. You won't really play a second hand if you win the first, but that doesn't change the odds — win-win and win-loss represent 50 percent of the possibilities, and you have a 50% probability of ending the contest in one hand.

Of those four possible two-decision outcomes, only one results in your brother-in-law winning overall. He had a 1 in 4 chance in winning.

With two trials, we've just seen there are four possible outcomes — 2 x 2. For four hands, raise 2 to the power of 4. There are 2 x 2 x 2 x 2 — 16 possible outcomes. Only one of those is win-win-win-win, so to answer your second question, your chances of winning four hands in a row were 1 in 16.

You didn't ask, but let's figure the chances of winning 10 trials in a row and ending the contest quickly. That would be 1 in 2 to the power of 10, which is 1 in 1,024.

Q. Is there any advantage to the player in getting free play instead of cash? More and more, the casinos I go to are giving rewards as free play,

A. If the amounts are equal, obviously I'd rather have cash than free play. However, if the knowledge that the customer will have to wager amounts equal to the free play before cashing out enables the casino to give larger rewards, that's a plus for the player.

I, too, find that most of the offers I receive by direct mail are for free play rather than vouchers redeemable for cash. Many casinos now have systems in place where rewards earned during play are downloadable right at the games, augmenting or replacing the old system of going to the player rewards booth for a voucher redeemable for cash at the cashiers' cage.

It's a trend that is bound to continue as free play is better than cash in assuring operators that rewards will go back into the games instead of walking out the door unplayed.

Q. I went to Las Vegas with some buddies, and one of them was a big video poker guy. He showed us some 10/7 Double Bonus Poker machines, and said that they paid more than 100% if you know how to play them right. I guess you have to be an expert, because I sure didn't get 100 percent, although we didn't do too bad.

He also said that even experts lose at more sessions than they win. How can you get a more than 100-percent payoff if you lose more than you win?

A. Full-pay Double Bonus Poker with a 10/7/5 pay table — meaning it pays 10 for 1 on full houses, 7 for 1 on flushes and 5 for 1 on straights — does indeed pay 100.17% with expert play. Expert strategy on the game is difficult, and few players actually beat the machines.

That better-than-100% payback includes a normal share of royal flushes, and with expert play, royals turn up an average of only once in 48,048 hands in this game. They account for 1.67% of the overall return in 10/7/5 Double Bonus, so in sessions when there is no royal flush, average return with expert play is only 98.5%.

Experts lose money on most sessions, but make it back — and more — in sessions that include royals.