1. Games of probability What are my chances?

2. Roll a single die (6 faces). –What is the probability of each number showing on top? Activity 1: Simple probability: Number on top Probability 1? 2? 3? 4? 5? 6? Number on top Probability 11/6 2 3 4 5 6 Assume the die is fair

3. Roll two dice. Sum of two diceProbability 2 1/11 (?) 3 4 5 6 7 8 9 10 11 12 Sum of two dice Can be done by:Probability 2(1, 1)1/36 3(1, 2) (2, 1)2/36 4(1, 3) (2, 2) (3, 1)3/36 5(1, 4) (2, 3) (3, 2) (4, 1)4/36 6(1, 5) (2, 4) (3, 3) (4, 2) (5, 1)5/36 7(1, 6) (2, 5) (3, 4) (4, 3) (5, 2) (6, 1)6/36 8(2, 6) (3, 5) (4, 4) (5, 3) (6, 2)5/36 9(3, 6) (4, 5) (5, 4) (6, 3)4/36 10(4, 6) (5, 5) (6, 4)3/36 11(5, 6) (6, 5)2/36 12(6, 6)1/36

4. Roll a die and toss a coin: –What is the probability of getting a “3 and Tail” ? die Coin Probability 1 H 1T 2H 2T 3H 3T? 4H 4T 5H 5T 6H 6T dieCoinProbability 1H1/12 1T 2H 2T 3H 3T 4H 4T 5H 5T 6H 6T Probability of getting a 3 on the die = 1/6. Probability of getting a tail on the coin = 1/2 Since the outcomes of the coin toss and the die rolling are independent, the join probability of getting a 3 AND a tail is (1/6)*(1/2)=1/12 Activity 2: Independence of two trials:

5. What if the coin is not fair? Assume the odds of getting the tail from the coin is 1/3, head is 2/3. –What is the probability of getting a “3 and Tail” now ? Answer: 1/6 * 1/3 = 1/18 dieCoinProbability 1H2/18 1T1/18 2H2/18 2T1/18 3H2/18 3T1/18 4H2/18 4T1/18 5H2/18 5T1/18 6H2/18 6T1/18

6. 1.Toss a coin. Each time it’s head, you win $1, each time it’s tail, you lose $1. 2.Roll two dice. Each time it’s 7, you win $4, otherwise you lose $1. 3.Roll two dice. Each time it’s 7, you win $5, otherwise you lose $1. 4.Roll two dice. Each time it’s 7, you win $6, otherwise you lose $1. Activity 3: Who is the winner? Even Loser Winner

7. 3 piles of cards. 2 cards in each pile: –Pile 1: ♥K and ♥K –Pile 2: ♥K and ♠K –Pile 3: ♠K and ♠K We don’t know which pile is which. Randomly pick one card from one pile. If the card we pick is ♥K, what is the odds that the other card in the pile is also ♥K? Let’s do an experiment! Activity 4: Don’t be fooled

8. Ways to pick ♥K : –if we happen to pick a card from pile 1: either card will do. –If we happen to pick a card from pile 2: only one card will do. –If we happen to pick a card from pile 3: no card will do. Probability of picking ♥K : (1/3)*(1)+(1/3)*(1/2)+(1/3)*0=1/2 Probability of picking a pile which has two ♥K: 1/3 So, knowing one card is ♥K, the probability of the other one is also ♥K is (1/3)/(1/2)=2/3

9. NO! Will you be a winner if you play this game? –Each time when ♠K is picked, no win, no lose. –Each time when ♥K is picked, you win $3 if the other card is ♠K. –Each time when ♥K is picked, you lose $2 if the other card is ♥K. You win $3 when you win, you only lose $2 when you lose…. Do you think you can make money by playing on? DON’T BE FOOLED!

10. Most of the gambling games are like this example – The odds are not in favor of the player. Use the concept of probability can help you determine whether a decision is good or bad – such as making investments. Don’t gamble – unless your math tells you that you can win.