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1 Unit 6: Probability Marius Ionescu 10/06/2011 Marius Ionescu () Unit 6: Probability 10/06/ / 22

2 Chapter 13: What is a probability Denition The probability that an event happens is the percentage of the time it happens when the same process is repeated many times in identical conditions. Marius Ionescu () Unit 6: Probability 10/06/ / 22

3 Examples Examples Marius Ionescu () Unit 6: Probability 10/06/ / 22

4 Examples Examples Coin toss Marius Ionescu () Unit 6: Probability 10/06/ / 22

5 Examples Examples Coin toss Die roll Marius Ionescu () Unit 6: Probability 10/06/ / 22

6 Examples Examples Coin toss Die roll Drawing a card from a pile Marius Ionescu () Unit 6: Probability 10/06/ / 22

7 Examples Examples Coin toss Die roll Drawing a card from a pile Picking a ball from a box (lottery) Marius Ionescu () Unit 6: Probability 10/06/ / 22

8 Examples Examples Coin toss Die roll Drawing a card from a pile Picking a ball from a box (lottery) Flipping a coin 1000 times, how many tails do you expect? Marius Ionescu () Unit 6: Probability 10/06/ / 22

9 Examples Examples Coin toss Die roll Drawing a card from a pile Picking a ball from a box (lottery) Flipping a coin 1000 times, how many tails do you expect? Rolling a die 6000 times, how many 2's do you expect? Marius Ionescu () Unit 6: Probability 10/06/ / 22

10 Examples Examples Coin toss Die roll Drawing a card from a pile Picking a ball from a box (lottery) Flipping a coin 1000 times, how many tails do you expect? Rolling a die 6000 times, how many 2's do you expect? Rolling a die 6000 times, how many 1,3,4,5, and 6s do you expect? Marius Ionescu () Unit 6: Probability 10/06/ / 22

11 Rules Fact (Rules of probability) Marius Ionescu () Unit 6: Probability 10/06/ / 22

12 Rules Fact (Rules of probability) 1 Probabilities must be between 0% and 100% Marius Ionescu () Unit 6: Probability 10/06/ / 22

13 Rules Fact (Rules of probability) 1 Probabilities must be between 0% and 100% 2 Prob (event A happens)=1 - Prob (event A does not happen) Marius Ionescu () Unit 6: Probability 10/06/ / 22

14 Rules Fact (Rules of probability) 1 Probabilities must be between 0% and 100% 2 Prob (event A happens)=1 - Prob (event A does not happen) 3 Prob(X ) = # of ways X can occur total number of outcomes (equally likely). Marius Ionescu () Unit 6: Probability 10/06/ / 22

15 Conditional probabilities Denition The probability of X given that Y happened is P(X Y ) = # of ways X and Y happen. # of ways Y occurs Marius Ionescu () Unit 6: Probability 10/06/ / 22

16 Example Example Suppose that we have an urn containing 3 red balls and 4 green balls. Marius Ionescu () Unit 6: Probability 10/06/ / 22

17 Example Example Suppose that we have an urn containing 3 red balls and 4 green balls. What is the probability that you extract one red ball? Marius Ionescu () Unit 6: Probability 10/06/ / 22

18 Example Example Suppose that we have an urn containing 3 red balls and 4 green balls. What is the probability that you extract one red ball? What is the probability that you extract one red ball if you extract rst a green ball? Marius Ionescu () Unit 6: Probability 10/06/ / 22

19 Independent Events Denition We say that X and Y are independent if P(Y X ) = P(Y ). Marius Ionescu () Unit 6: Probability 10/06/ / 22

20 Example Example Suppose that you are ipping a coin 5 times. Marius Ionescu () Unit 6: Probability 10/06/ / 22

21 Example Example Suppose that you are ipping a coin 5 times. What is the probability that the 4 th trial is head? Marius Ionescu () Unit 6: Probability 10/06/ / 22

22 Example Example Suppose that you are ipping a coin 5 times. What is the probability that the 4 th trial is head? What if you know that the rst four trials were all tails? Marius Ionescu () Unit 6: Probability 10/06/ / 22

23 Chapter 14: AND Rule Fact (AND Rule) P(X and Y ) = P(X ) P(Y X ) = P(Y ) P(X Y ) Marius Ionescu () Unit 6: Probability 10/06/ / 22

24 Examples Examples Find the probability of each of the following events: Marius Ionescu () Unit 6: Probability 10/06/ / 22

25 Examples Examples Find the probability of each of the following events: Drawing the ace of clubs and then the two of diamonds. Marius Ionescu () Unit 6: Probability 10/06/ / 22

26 Examples Examples Find the probability of each of the following events: Drawing the ace of clubs and then the two of diamonds. Drawing a club and then a heart. Marius Ionescu () Unit 6: Probability 10/06/ / 22

27 Examples Examples Find the probability of each of the following events: Drawing the ace of clubs and then the two of diamonds. Drawing a club and then a heart. Drawing ve clubs Marius Ionescu () Unit 6: Probability 10/06/ / 22

28 Examples Examples Find the probability of each of the following events: Drawing the ace of clubs and then the two of diamonds. Drawing a club and then a heart. Drawing ve clubs Rolling 2 sixes Marius Ionescu () Unit 6: Probability 10/06/ / 22

29 The paradox of the Chevalier de Méré Example The Chevalier de Méré made the following computations: Rolling a die In one roll of a die, I have 1/6 of a chance to get an ace (a one). So in 4 rolls, I have 4 1/6 = 2/3 of a chance to get at least one ace (a least a one) Marius Ionescu () Unit 6: Probability 10/06/ / 22

30 The paradox of the Chevalier de Méré Example The Chevalier de Méré made the following computations: Rolling a die In one roll of a die, I have 1/6 of a chance to get an ace (a one). So in 4 rolls, I have 4 1/6 = 2/3 of a chance to get at least one ace (a least a one) Rolling a pair of dice In one roll of a pair of dice, I have 1/36 of a chance to get a double-ace (two ones). So in 24 rolls, I must have 24 1/36 = 2/3 of a chance to get at least one double-ace. Marius Ionescu () Unit 6: Probability 10/06/ / 22

31 The paradox of the Chevalier de Méré Example The Chevalier de Méré made the following computations: Rolling a die In one roll of a die, I have 1/6 of a chance to get an ace (a one). So in 4 rolls, I have 4 1/6 = 2/3 of a chance to get at least one ace (a least a one) Rolling a pair of dice In one roll of a pair of dice, I have 1/36 of a chance to get a double-ace (two ones). So in 24 rolls, I must have 24 1/36 = 2/3 of a chance to get at least one double-ace. Do you agree with his statements? Marius Ionescu () Unit 6: Probability 10/06/ / 22

32 OR Rule Fact (OR Rule) P(X or Y ) = P(X ) + P(Y ) P(X and Y ). Marius Ionescu () Unit 6: Probability 10/06/ / 22

33 Examples Examples Find the probability of the following events Marius Ionescu () Unit 6: Probability 10/06/ / 22

34 Examples Examples Find the probability of the following events You draw a Heart or a Queen. Marius Ionescu () Unit 6: Probability 10/06/ / 22

35 Examples Examples Find the probability of the following events You draw a Heart or a Queen. You pick a two or a ve. Marius Ionescu () Unit 6: Probability 10/06/ / 22

36 Examples Examples Find the probability of the following events You draw a Heart or a Queen. You pick a two or a ve. You pick a red card or a jack. Marius Ionescu () Unit 6: Probability 10/06/ / 22

37 Examples Examples Find the probability of the following events You draw a Heart or a Queen. You pick a two or a ve. You pick a red card or a jack. You pick an Ace and then a King. Marius Ionescu () Unit 6: Probability 10/06/ / 22

38 Examples Examples Find the probability of the following events You draw a Heart or a Queen. You pick a two or a ve. You pick a red card or a jack. You pick an Ace and then a King. You roll 0 sixes in 5 rolls Marius Ionescu () Unit 6: Probability 10/06/ / 22

39 Mutually exclusive events Denition We say that X and Y are mutually exclusive if P(X and Y ) = 0. Marius Ionescu () Unit 6: Probability 10/06/ / 22

40 Example Example Suppose that you role a die. What is the probability that you get a 2 and a 3? Marius Ionescu () Unit 6: Probability 10/06/ / 22

41 What if it is hard to nd the probability of an event? Fact Keep in mind the following: If the probability that an event A happens seems hard to compute, then try to nd the probability that A does not happen, that is, 1 P(A). Marius Ionescu () Unit 6: Probability 10/06/ / 22

42 Chapter 15: Binomial Formula Example Marius Ionescu () Unit 6: Probability 10/06/ / 22

43 Chapter 15: Binomial Formula Example How many ways can we arrange 3 books on a table? Marius Ionescu () Unit 6: Probability 10/06/ / 22

44 Chapter 15: Binomial Formula Example How many ways can we arrange 3 books on a table? How many ways can we arrange 5 balls with 3 red and 2 green? Marius Ionescu () Unit 6: Probability 10/06/ / 22

45 Chapter 15: Binomial Formula Example How many ways can we arrange 3 books on a table? How many ways can we arrange 5 balls with 3 red and 2 green? How many ways can a class of 21 students be divided into a group of 10 and another group of 11? Marius Ionescu () Unit 6: Probability 10/06/ / 22

46 Factorials and the binomial coeecient Denition Marius Ionescu () Unit 6: Probability 10/06/ / 22

47 Factorials and the binomial coeecient Denition n-factorial is n! = n(n 1) Marius Ionescu () Unit 6: Probability 10/06/ / 22

48 Factorials and the binomial coeecient Denition n-factorial is The binomial coecient n! = n(n 1) n! k!(n k)! Marius Ionescu () Unit 6: Probability 10/06/ / 22

49 More examples Example Marius Ionescu () Unit 6: Probability 10/06/ / 22

50 More examples Example How many ways can 21 dice come up with 10 sixes? Marius Ionescu () Unit 6: Probability 10/06/ / 22

51 More examples Example How many ways can 21 dice come up with 10 sixes? How many way can 4 dice come up with 4 sixes? Marius Ionescu () Unit 6: Probability 10/06/ / 22

52 Binomial Formula Fact If p is the probability of success on each trial and we do n independent trials, then P(k successes out of n trials) = n! k!(n k)! pk (1 p) n k. Marius Ionescu () Unit 6: Probability 10/06/ / 22

53 Example Example Marius Ionescu () Unit 6: Probability 10/06/ / 22

54 Example Example When drawing 6 balls with replacement from a box with 8 green and 5 red balls, what is the probability of getting 3 red and 3 green? Marius Ionescu () Unit 6: Probability 10/06/ / 22

55 Example Example When drawing 6 balls with replacement from a box with 8 green and 5 red balls, what is the probability of getting 3 red and 3 green? If 10 draws are made with replacement, what is the probability that two or less balls are red? Marius Ionescu () Unit 6: Probability 10/06/ / 22

56 Example Example Marius Ionescu () Unit 6: Probability 10/06/ / 22

57 Example Example What is the probability of rolling 3 sixes on 6 rolls? Marius Ionescu () Unit 6: Probability 10/06/ / 22

58 Example Example What is the probability of rolling 3 sixes on 6 rolls? What is the probability of rolling at least 4 sixes on 6 rolls? Marius Ionescu () Unit 6: Probability 10/06/ / 22

### Unit 6: Probability. Marius Ionescu 10/06/2011. Marius Ionescu () Unit 6: Probability 10/06/ / 22

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