1. 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?

2. 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?

3. 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

4. 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

5. g -5 -3 5 P(G=g) 0.25 𝐸 𝐺 = 𝑔.𝑃(𝐺=𝑔) = -0.755
P(G=g)
0.25
The work out the Expected Value of this distribution:
𝐸 𝐺 = 𝑔.𝑃(𝐺=𝑔)
= -0.75

6. 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

7. 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.

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