Using Probabilities to Make Fair Decisions Section 4.9

Objectives Compare different probabilities. Deciding if probabilities are fair.

Example; Make Fair Decisions Two teams decided to play cricket. They want to decide who bats first. Robert and David are the team captains. So they have two suggestions to decide who bats first. Decide whether the suggestions are fair ways to make the decision. Robert says, “We can flip a coin, heads my bats first. If it is tails David’s team bats first.” David says, “We can roll a single die. If it lands on either 1,2, or 3, my team bats first. If the roll is a 4, 5, or 6 then Robert’s team bats first. Which is the better option? Explain.

Example–Solution Robert’s Suggestion… P(Head) = 1/2 = Robert’s Team P(Tail) = 1/2 = David’s Team Robert’s Team = David’s Team Robert’s suggestion is fair because they have the same probability of getting to bat first.

Example–Solution David’s Suggestion… P(1, 2, or 3) = 1/2 = David’s Team P(4, 5, or 6) = 1/2 = Robert’s Team David’s Team = Robert’s Team David’s suggestion is fair because they have the same probability of getting to bat first.

Probability of the Number Example; Make Fair Decisions Some of Allan’s friends are playing with dice. They decide to roll a die. If it lands on a 1 then he wins $5. If it lands on a 6 then he wins $10. If it lands on any other number he loses $3. Is the game fair? Explain. Solution: Number Rolled Number 1 Number 6 Number (2, 3, 4, 5) Probability of the Number Amount of Money ($) Pay Out 1/6 1/6 2/3 $5 $10 -$3 5/6 10/6 -12/6

Example–Solution Total Payout = 5/6 + 10/6 – 12/6 Total Payout = 1/2 1/2 ≠ 0 Since the total average payout is not zero. The game is not fair because the probabilities of winning are not the same for each pay out. Alan is more likely to lose money than win money.

P(1) = 2/8 = 1/4 P(2) = 2/8 = 1/4 P(3) = 2/8 = 1/4 P(4) = 2/8 = 1/4 Example; Make Fair Decisions Caleb made a spinner. Is this spinner fair? Explain. Solution: 1 4 2 3 P(1) = 2/8 = 1/4 P(2) = 2/8 = 1/4 P(3) = 2/8 = 1/4 P(4) = 2/8 = 1/4 Since the spinner has 8 equal sections and each number is represented an equal number of times this spinner is fair.

Example; Make Fair Decisions Charlie and Molly want to play basketball. To decide who will take the ball first they make two spinners. Spinner 1 Spinner 2 If the spinner lands on Blue Charlie gets the ball. If the spinner lands on Yellow Molly gets the ball first.   Which spinner is fair if either? Explain. B Y Blue Yellow

Example–Solution Spinner 1… P(Blue) = 1/2 = Charlie P(Yellow) = 1/2 = Molly Charlie = Molly Spinner 1 is fair because both players have and equal probability of getting the ball first.

Example–Solution Spinner 2… P(Blue) = 3/8 = Charlie P(Yellow) = 5/8 = Molly Charlie ≠ Molly Spinner 2 is not fair because both players do not have and equal probability of getting the ball first.

How to Make Fair Decisions 1) Find probabilities of each situation 2) Are the probabilities the same 3) ‘Same’ = ‘Fair’ or ‘Not Same’ = ‘Not Fair’ 4) If we are looking at expected outcomes, we want the balance to be zero for fairness.

4.9 Using Probabilities to Make Fair Decisions Summarize Notes Read section 4.9 Homework Worksheet