Gain Expected Gain means who much – on average – you would expect to win if you played a game. Example: Mark plays a game (for free). He throws a coin. If it’s a Head he wins $1. If he throws a Tail he loses $1. What is his expected gain?

In a new game: Mark pays $1 to enter. If he gets a head he wins $3. If he gets a tail he loses $2. What is his expected gain?

In an attempt to create a really complicated game, and make some money, Mark makes a spinner. He charges $5 to play the game. If a player spins a 1, the player loses the money they paid. If they spin a 2, they get $2 back. If they spin a 3, they get their money back. If they spin a 4, they get their money back and $5. 4   3 2 1

g -5 -3 5 P(G=g) 0.25 We make a distribution table for the Gains: If a player spins a 1, the player loses the money they paid. If they spin a 2, they get $2 back. If they spin a 3, they get their money back. If they spin a 4, they get their money back and $5. We make a distribution table for the Gains: g -5 -3 5 P(G=g) 0.25

g -5 -3 5 P(G=g) 0.25 𝐸 𝐺 = 𝑔.𝑃(𝐺=𝑔) = -0.75 5 P(G=g) 0.25 The work out the Expected Value of this distribution: 𝐸 𝐺 = 𝑔.𝑃(𝐺=𝑔) = -0.75

This means that on average a player will lose $0 This means that on average a player will lose $0.75 for every game they play. Now try the Learning Workbook p. 6 Exercise B, Question 1

Fair: A game is ‘fair” if the expected gain is 0. Two new words: Winnings: Similar to “gain”, but ignores the cost of playing the game. Fair: A game is ‘fair” if the expected gain is 0.

Now do the Learning Workbook p. 6 – 8 Exercise B.