Ask The Wizard #279

On average, how many trials will it take in a 50/50 game to lose two in a row? How about 3, 4, n in a row?


Let's solve for the two loss case first.

Let x be the expected number of future flips starting from the beginning or after any win.
Let y be the expected number of future flips after one loss.

We can set up the following two equations:

(1) x = 1 + .5x + .5y

The one represents that the player must flip the coin to change states. There is a 50% chance of a win, remaining in state x. There is a 50% chance of a loss, advancing to state y.

(2) y = 1 + .5x

From state y again, the 1 represents the flip at that point. There is a 50% chance of a win, reverting to state x. There is a 50% chance of a loss, ending the game, necessitating no additional flips, so it is an implied 0.5*0.

Multiply both equations by 2 and reorder to get:
(3) x - y =2
(4) -x + 2y = 2

Add the two equations to get:

(5) y = 4

Plug that into any equation from (1) to (4) and get x=6.

For the three loss case, define the three possible states as:

Let x be the expected number of future flips starting from the beginning or after any win.
Let y be the expected number of future flips after one loss.
Let z be the expected number of future flips after two losses.

The initial equations are:

x = 1 + .5x + .5y
y = 1 + .5x + .5z
z = 1 + .5x

We can set up the initial states in matrix form as:

0.5 -0.5 0 1
-0.5 1 -0.5 1
-0.5 0 1 1

If you remember your matrix algebra, we can solve for x as determinant(A)/determinant(B) where

A =

1 -0.5 0
1 1 -0.5
1 0 1

B =

0.5 -0.5 0
-0.5 1 -0.5
-0.5 0 1
0.5 -0.5 0
-0.5 1 -0.5
-0.5 0 1

Excel has a handy determinant function: =mdeterm(range). In this case x = mdeterm(matrix A)/mdeterm(matrix B) = 1.75/0.125 = 14.

We can use recursion for additional consecutive losses. Let's consider 4. We know from above it will take 14 flips on average to get 3 losses in a row. At that point the coin will be flipped again, with a 50% chance of starting over. So:

x = 14 + 1 + x/2
x/2 = 15
x = 30

In other words, add one to the previous answer and then double.

It is not difficult to see the pattern. The expected number of flips to get n losses in a row is 2n+1-2.

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

In your "Ask the Wizard" column #277, you were asked about the oldest casino game patent. Your answer mentioned Caribbean Stud Poker. That was not the first casino patent granted, although it might be the first patent on a successful game. I checked Google patents (believe it or not but the USPTO site does not go before 1976), and the earliest patent I found on any casino-type game is a patent for a game device issued in 1898.

Jon M. from Philadelphia, PA

Thanks. I can't tell what that is a patent for, but interesting to know that the business of inventing casino games goes that far back.

Doyle Brunson famously won the Main Event in the World Series of Poker in both 1976 and 1977. Each time he held 10-2 as hole cards, and both times he made a full house on the river. What are the odds of that?

Jonathan F.

Given two cards of different ranks, the probability of making a full house are 1 in 121.6. The odds of making it on the river are 1 in 207.

The odds of making such a hand on the river two out of two times is 1 in 43,006.

The odds of this happening with the same two starting cards, in rank only, are 1 in 3,564,161.

The odds of this happening with exactly 10-2 both times is 1 in 295,379,826.

What is supposed to happen in a poker game if a player dies mid-hand?


I asked a former Nevada gaming regulator and casino president about this. He said it would be treated as an "all in" kind of situation, much the same way an accidental disconnection is handled in Internet poker.

In other words, a side pot would be made of chips in the middle at the time of death. Then, any additional bets would be set aside in a separate pot. If the dead player had the highest hand, then he would win the side pot. Win or lose, any chips he had on the table after the hand would be set aside for the deceased's estate.

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

At the pai gow table, can I ask the dealer to review how I set my tiles to see if I did it the house way? How often is the house way the best way to play a hand?


As long as you are not slowing down the game, particularly when there are big bettors at the table, you can usually show how you would set your tiles and ask the dealer, "Is this how you would do it?" It will also depend on how patient the dealer is and/or whether the other players seem to object. One dealer I know didn't like to be asked because she said it confused her when she had to set her own hand. With any difficult game, if you're a beginner I would recommend trying to get a table to yourself the first time, so you don't inconvenience other players with a lot of questions.

Regarding the second question, if the player is going against the traditional house way, the house way will be correct 80.2% of the time. That other 19.8% is another reason why pai gow is such a difficult game to master.

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

In Dice Wars, what is the probability of success for any given number of attacking and defending dice? As an attacker, what ratio has the greatest expected gain?


For those unfamiliar with the game, both the attacker and defender will roll 1 to 8 dice, according to how many armies they each have at that point in a battle. The higher total shall win. A tie goes to the defender. If the attacker loses, he will still retain one army in the territory where he initiated the attack. For this reason, he must have at least two armies to attack, so if he wins one can inhabit the conquered territory and one can stay behind.

The following table shows the probability of an attacker victory according to all 64 combinations of total dice.

Probability of Attacker Win

Attacker Defender
1 Army 2 Armies 3 Armies 4 Armies 5 Armies 6 Armies 7 Armies 8 Armies
2 0.837963 0.443673 0.152006 0.035880 0.006105 0.000766 0.000071 0.000005
3 0.972994 0.778549 0.453575 0.191701 0.060713 0.014879 0.002890 0.000452
4 0.997299 0.939236 0.742831 0.459528 0.220442 0.083423 0.025450 0.006379
5 0.999850 0.987940 0.909347 0.718078 0.463654 0.242449 0.103626 0.036742
6 0.999996 0.998217 0.975300 0.883953 0.699616 0.466731 0.259984 0.121507
7 1.000000 0.999801 0.994663 0.961536 0.862377 0.685165 0.469139 0.274376
8 1.000000 0.999983 0.999069 0.989534 0.947731 0.843874 0.673456 0.471091

The next table shows the expected gain by the attacker, defined as pr(attacker wins)*(defender dice)+pr(defender wins)*(attacker dice -1). It shows the greatest expected gain is to attack with 8 against an opponent with 5.

Net Gain of Attacker Win

Attacker Defender
1 Army 2 Armies 3 Armies 4 Armies 5 Armies 6 Armies 7 Armies 8 Armies
2 0.675926 0.331019 -0.391976 -0.820600 -0.963370 -0.994638 -0.999432 -0.999955
3 0.918982 1.114196 0.267875 -0.849794 -1.575009 -1.880968 -1.973990 -1.995480
4 0.989196 1.696180 1.456986 0.216696 -1.236464 -2.249193 -2.745500 -2.929831
5 0.999250 1.927640 2.365429 1.744624 0.172886 -1.575510 -2.860114 -3.559096
6 0.999976 1.987519 2.802400 2.955577 1.996160 0.134041 -1.880192 -3.420409
7 1.000000 1.998408 2.951967 3.615360 3.486147 2.221980 0.098807 -2.158736
8 1.000000 1.999847 2.990690 3.884874 4.372772 3.970362 2.428384 0.066365

Did you see the Dilbert cartoon on Jan 23, 2012? Do you suppose Wally was playing pai gow (tiles) or pai gow poker?

Michael M. from Las Vegas

Yes! I loved it. I think Wally was playing tiles. Here are my reasons:

  • Wally looks like the kind of non-Asian player one would find at a tile table.
  • Dilbert is the scientific type, who normally are sticklers for using correct terminology. Calling pai gow poker "pai gow" is incorrect and lazy. I know most people do so anyway, but I have higher expectations for Dilbert.
  • In the second frame, Dilbert says that pai gow is "a difficult game to learn after a few adult beverages." Note that he said "learn," not "play." Pai gow poker is not that hard to learn. If you understand how to play poker, then pai gow poker can be easily explained in under a minute. Meanwhile, tiles is difficult to learn as well as play.
  • The cartoon came out on Chinese New Year. This may have been an inside joke.

In the unlikely event Scott Adams should read this, I would welcome a definitive answer.

This question was discussed on my forum at Wizard of Vegas.