Fair games

Rules for various games are given. Use expected value to help analyze the game. Notice that a "fair game" means all players' expected values are equal, or that the game's expected value is 0.

Roll 2d6 (2 dice with 6 sides).

• If sum is odd, player A gets 1 point.
• If sum is even, player B gets 2 points.

Is this game fair? If fair, explain why. If not, provide a scoring scheme to make the game fair.

Multiplication game

Roll 2d6.

• If product is even, player A scores 1 point.
• If product is odd, player B scores 3 points.

Is this game fair? If fair, explain why. If not, provide a scoring scheme to make the game fair.

"Get six" game

Roll 1d6.

• Get a six and win $4. • Get anything else and lose$1.

If 100 people play this game, how much money will the game makers (not the players) expect to make?

"Get six" game (redux)

Pay $1. Roll 1d6. • Get a six and win$4.
• Get anything else and gain nothing.

If 100 people play this game, how much money will the game makers (not the players) expect to make?

Deep Space D6

There are three possible attackers at the end of this round. One die will be rolled and the damage of each attacker is listed below the dice faces.

We can block one of the attackers using the green dice at the bottom. Which attacker should we block and why?

"Pick one" using two dice

Pick a number and roll 2d6.

• Picked number comes up twice, win $4. • Picked number comes up once, win$1.
• Picked number doesn't appear, lose $2. What are the expected earnings for the player of this game? (hint: it doesn't matter which number picked) Challenge: "Pick one" using three dice Pick a number and roll 3d6. • Picked number comes up thrice, win$9.
• Picked number comes up twice, win $4. • Picked number comes up once, win$1.
• Picked number doesn't appear, lose $2. How many outcomes are there? (hint: more than 18) What are the expected earnings for the player of this game? (hint: a weighted tree diagram is recommended because your number either appears, or doesn't) Challenge: WI Lottery Review the odds for Powerball (https://wilottery.com/games/powerball). A ticket costs$2. Assume the jackpot is \$250,000,000.

1. Recall that odds represent different events, not total outcomes. For example, 1:39 odds means there is 1 chance to win and 39 chances to lose. Thus the probably of winning is 1/40. What are all the probabilities of winning money in the PowerBall?
2. Based on your previous work, what are the expected earnings per ticket?