Dice, Cards and Darts

Probabilities in Games

Finding the Right Representation

Finding a winning strategy – playing the odds

• How do we represent probabilities?– Discuss

• Odds vs. Probabilities

Simple dice games

• Many dice games use throws of two, or more, dice and the sum of the scores.

• What are the most likely scores?– Experiment in class– Collect data

• Can we explain it?• Can we represent what is

going on? Excel: Dice patterns

What about nonstandard dice?

Let’s play!!

What did we find?

• Is there a pattern?

Excel: Dice patterns

• What would have happened with multiplication?

• What about other dice? – Create your own spreadsheets in

groups.

Variations to consider

• Suppose one conventional die and one with values 0, 1, 2 with probabilities 1/3, 1/2, 1/6

• What are values and probabilities of possible sums? Products?

• Can we modify our tables for this or other variations?

Cards – blackjack “score”

• Deal two cards from a standard deck

• Add values with the following rules:→Ace counts 1 or 11 at player’s option→2 – 10 count for face value→Jack, Queen, King all count 10

• What are the possible scores?

What is a good way of representing/showing this?

Cards – blackjack “score”

• Deal two cards from a standard deck

• Add values with the following rules:→Ace counts 1 or 11 at player’s option→2 – 10 count for face value→Jack, Queen, King all count 10

• Over 21 is “bust” so …When should you stand?

• What if you can see one of the dealers cards?

Technology?

• Excel spreadsheet can help compute odds– Scoring chart spreadsheet – exercise– Use experiment to verify/validate model– Test your strategy in actual play

• Download a program to play/deal/check your odds?– http://www.qfit.com/CVDPC.htm– http://www.s-a-g-e.com/blackjack_info/bjinfo.html – Game program: Sage Blackjack (shareware)

Darts?

• Strategy for maximum score– Ignore doubles, triples, bulls

• What about best chance to get a fixed score? – This is typically the way a game finishes

• Strategy depends on skill level– Conditional probabilities

Darts

• Why is a dart board arranged the way it is?

• Could it be designed better?

• What is your expected score if you aim at 20 each time?– How does this change for

different target locations on the board?

Who is the better hitter?

• Two baseball hitters, Jose and Brian have the batting averages 0.260 and 0.264 respectively. – Which is the better batter?

Hitter Batting Average

Jose 0.260

Brian 0.264

Who is the better hitter?• These two hitters have the batting

averages shown when facing left or right-handed pitchers. – Which is the better batter?– Are these statistics consistent?

Hitter Average v. lefties

Average v. righties

Jose 0.300 0.250

Brian 0.270 0.240