Fair Games, Odds, and Counting

Sum Game

Fair Game versus Fair Play

Fair Game—when everyone plays the game fairly by not cheating, playing by the rules, etc..

Fair Play—when each player of the game has an equal chance of winning the game.

The theoretical probability of each player winning a fair game is 1/2.

Tree Diagram—count success over total

P(Player 1) = 12/36 = 1/3

P(Player 2) = 24/36 = 2/3

Chips

B

A

Outcomes

AAB

AAC

ACB

ACC

BAB

BAC

BCB

BCC

P(Player 1) = 6/8 = 3/4

P(Player 2) = 2/8 = 1/4

Def: __Odds__—the ratio of the probability that an event
will occur to the probability that the event will not occur.

_{}

Example: What are the odds of getting a “z” on this spinner?

_{}= _{} = 1/2 or 1:2

Example:

Experiment: Flip a coin twice

What are the odds of getting at least on head?

_{}

P(at least one head) = 3/4 Odds 3:1

Notice any patterns?

Odds—successes:nonsuccesses

Probability--_{}

- Sum Game

Odds
of Player 2 winning are 2:1 _{}

- Chips Game

Odds
of Player 2 winning are 1:3 _{}

- Roll a die. What are the odds of getting a number less than 6?

_{} Odds are 5:1

- Suppose P(A) = 3/7. What are the odds of A?

3:4

- Suppose the odds of B are 5:3. What is the P(B)?

_{}

Counting

Suppose you were to spin the two spinners below. Draw a tree diagram to find how many outcomes are possible.

12 outcomes

Notice pattern: Why does this happen?

Counting Rule:

- Number
of outcomes for 4 dice:
_{} - Flip a
coin, roll a die:
_{} - Spin a
spinner 4 times:
_{} - License plates

· First 2 characters are numbers

· Middle 3 characters are letters

· Last character is a number

How many license plates are possible?

_{}

- Same as number 4 but no repetition

_{}