Monday 23 June 2014

Gambling and Expected Value: Crown and Anchor

In this post on Gambling and Expected Value, we look at the game Crown and Anchor.
Click here to find similar posts on other lotteries and games of chance.

How the Game Works

The game Crown and Anchor is a dice game that historically was often played by British sailors. Instead of ordinary dice, each face has one of six symbols: crown, anchor, spade, diamond, club, and heart.

 A Crown and Anchor mat with dice.

Players can bet on one or more symbols coming up and then roll three dice. If the symbol you bet on comes up, you win. The payout depends on how many of the dice are showing the symbol.

There is also a variant of the game played with a wheel instead of dice. The wheel is divided into 216 segments, where each segment shows one the 216 possible ways to choose from the six symbols three times. The banker in this case spins the wheel and the winner is determined based on which segment is showing when the wheel stops.

Probabilities and Prizes

If the winning symbol appears on one of the three dice, the payout is 1:1 on the wager. If two of the three dice are showing the winning symbol, it's a 2:1 payout. Likewise, if all three dice are showing the symbol, it's a 3:1 payout. So there are four possible outcomes for a wager (loss, 1:1 payout, 2:1 payout, and 3:1 payout). Each die has 6 symbols, so there are 6 * 6 * 6 = 216 possible combinations of symbols when three dice are rolled. The probability of losing is 5/6 * 5/6 * 5/6 = 125/216. That leaves 91 possible ways to win (approximately a 42% chance of winning something). Only one of those combinations would be all three symbols matching the wager. Therefore, there's a 1 in 216 chance of winning the 3:1 payout. There are 15 possible ways to get exactly two matching symbols, so there is a 15 in 216 = 5 in 72 chance of winning the 2:1 payout. Finally, there are 75 ways to get exactly one matching symbol, corresponding to a 75 in 216 = 25 in 72 chance of winning the 1:1 payout.

Expected Value

The expected value of Crown and Anchor is about -\$0.079 per dollar wagered, which was calculated as follows:

This means that, on average, you lose nearly 8 cents of every dollar wagered. A lot better than lotteries or Monte or 50/50 draws, allowing you to play longer with your money. It's much more comparable to casino games, though there are other options with even better expected value.

1. I don't have a massive amount of experience with statistics, but that second last equation doesn't seem to take into account the fact that you bet \$1 regardless of whether you win or lose. To my mind it should be:
EV=1/216*3+5/72*2+25/72-1

1. If you win, your wagered dollar remains on the table and the payout is added to it. The alternate way of expressing the expected value would be:
EV = (1/216)(3+1) + (5/72)(2+1) + (25/72)(1+1) - 1 = -17/216.

2. This comment has been removed by the author.

3. Because there are 6 symbols and each has as group of 3 on the wheel would that not make the odds of a 3:1 payout 6 in 216 or 1:36? Not 1:216?

1. There is a 1:36 chance that the wheel lands on one of the spots that pays out 3:1, and a 1:6 chance that you happened to pick the right symbol. 1/36 * 1/6 = 1/216.

4. Columbus cruised for America. His bet: that he would wind up in the Far East. He didn't get was he was going for - yet the bet satisfied for the Europeans. bitcoin blackjack casino

5. This comment has been removed by the author.

6. Could potentially be at the same time a fairly fantastic post that marilyn and i clearly loved comprehending. Usually not regularly of which see the the chances to buy a device. 토토사이트

7. Pretty good post. I just stumbled upon your blog and wanted to say that I have really enjoyed reading your blog posts. Any way I'll be subscribing to your feed and I hope you post again soon. Big thanks for the useful info. mybestpoker

8. Online casino gambling can be very thrilling. The fun never stops when gambling online. The rush you experience when you win something is matchless. The fun and thrill can be increased further if you follow certain unwritten tips while gambling online. These tips will help in making the online gambling experience an enjoyable one. It will minimize the possibility of you shedding tears of frustration and failure. Let us call these tips the do's of online casino gambling.Judi Slot

9. Situs Agen qq online Terbaik yaitu SBOBETPK Dengan pelayanan terbaik se jakarta
Membuat Para member menjadi seperti keluarga besar, dna keuntungan yang bisa di dapatkan lainnya adalah
Sekarang SBOBETPk bisa Deposit Poker Via Pulsa Telkomsel dan Xl

10. SBOBETPK sebagai salah satu situs judi online poker terpercaya senantiasa memberikan promo promo bonus terpercaya dan 100% terbayarkan. Bonus tersebut diantaranya Bonus Turn Over Mingguan 0.5%, Bonus Refferal Mingguan 20% dan Bonus di poker online dan Situs Poker Online Terpopuler. Dengan adanya bonus yang bervariasi tentu saja memberikan peluang keuntungan lebih besar untuk setiap member.

11. Thank you because you have been willing to share information with us. we will always appreciate all you have done here because I know you are very concerned with our. DNA Poker

12. Thanks for a very interesting blog. What else may I get that kind of info written in such a perfect approach? I’ve a undertaking that I am simply now operating on, and I have been at the look out for such info. Agen Poker

13. Agen Depsoit Pulsa Potongan Termurah

Villaslots Agen Slots Online Terpercaya yang memberikan promo menarik dan pelayanan terbaik
Situs Judi Online yang menyediakan Deposit Pulsa Slots dengan Potongan Termurah

PROMO :
- Bonus New Member 100%
- Bonus Deposit Harian 100%
- Bonus Rollingan Slots 0.5%

villaslots
deposit pulsa potongan termurah
agen slots online terpercaya

Villaslots is a Trusted Online Slots Agent

14. AGEN SLOTS DEPOSIT PULSA POTONGAN TERMURAH

villaslots Agen Slots Online Terpercaya yang memberikan promo menarik dan pelayanan terbaik
Situs Judi Online yang menyediakan Deposit Pulsa Slots dengan Potongan Termurah

PROMO :
- Bonus New Member 100%
- Bonus Deposit Harian 50%
- Bonus Rollingan Slots 0.5%

WhatsApp : +6282284941566

villaslots
villa slots
deposit pulsa
deposit pulsa slots
deposit pulsa poker
deposit pulsa joker123
deposit pulsa tanpa potongan
deposit pulsa potongan termurah
agen deposit pulsa villa slots
deposit pulsa terepercaya
agen slots online terpercaya

Villaslots is a Trusted Online Slots Agent

15. AGEN SLOTS DEPOSIT PULSA POTONGAN TERMURAH

villaslots Agen Slots Online Terpercaya yang memberikan promo menarik dan pelayanan terbaik
Situs Judi Online yang menyediakan Deposit Pulsa Slots dengan Potongan Termurah

PROMO :
- Bonus New Member 100%
- Bonus Deposit Harian 50%
- Bonus Rollingan Slots 0.5%

WhatsApp : +6282284941566

villaslots
villa slots
deposit pulsa
deposit pulsa slots
deposit pulsa poker
deposit pulsa joker123
deposit pulsa tanpa potongan
deposit pulsa potongan termurah
agen deposit pulsa villa slots
deposit pulsa terepercaya
agen slots online terpercaya

Villaslots is a Trusted Online Slots Agent

16. I am incapable of reading articles online very often, but I’m happy I did today. It is very well written, and your points are well-expressed. I request you warmly, please, don’t ever stop writing. 로투스게임

17. You have a real ability for writing unique content. I like how you think and the way you represent your views in this article. I agree with your way of thinking. Thank you for sharing. wm

18. I really thank you for the valuable info on this great subject and look forward to more great posts. Thanks a lot for enjoying this beauty article with me. I am appreciating it very much! Looking forward to another great article. Good luck to the author! All the best! เกมยิงปลา

19. What a fantabulous post this has been. Never seen this kind of useful post. I am grateful to you and expect more number of posts like these. Thank you very much. แทงหวย