Microsoft produces a New operating system on a disk. There is 0 Microsoft produces a New operating system on a disk. There is 0.12 probability that it will contain just one bug in the programming. There is a 0.08 probability that there may be two bugs, and 0.04 probability that there are 3+ bugs. What is the probability that you have a bug free operating system? What is the probability that you have at most 2 bugs in the system? If you buy TWO disks what is the probability that both disks have 3+ bugs? If you buy TWO disks what is the probability that at least ONE has a bug? [Hint: P(At Least One) = 1 –P(None)] WARM UP 1 – (.12+.08+.04) = 0.76 .76+.12+.08 = 0.96 (.04)(.04) = 0.0016 No Bugs on TWO disks = (.76)(.76)=.5776 P(At least One)= 1 – .5776 = 0.4224

http://goo.gl/41DaSJ

A B C D VENN DIAGRAM 55 – 12 = 43 125 – 12 = 113 = 12 1600 – (113+12+43) = D A = 125 students in Calculus; B = 55 students in Statistics; C = 12 students in both Calc. and Stats; & D = Students in Neither courses. There are 1600 students at the school. Are Events A and B disjoint? How many Students represents (A and B) = ? How many Students represents Ac = ? How many Students represents D = ? NO! 12 1475 1432

Are mutually exclusive events Independent or Dependent? Independent Events: Two events in which the occurrence of one event has NO EFFECT on the other. P(A ∩ B) = P(A)·P(B) Mutually Exclusive Events are disjoint events. Two events can NEVER occur at the same time. This means that: P(A ∩ B) = 0 Are mutually exclusive events Independent or Dependent?

a.) If X and Y are Mutually Exclusive Find the Probability of X or Y EXAMPLE 1: Given that P(X) = 0.25 and P(Y) = 0.36 P(Z) = 0.15 find: a.) If X and Y are Mutually Exclusive Find the Probability of X or Y b.) If X and Y are Mutually Exclusive Find the Probability of X and Y c.) If X and Y are Independent Find the Probability of X and Y d.) If Y and Z are Independent and X is M.E. to Y and Z, Find the Probability of X or (Y and NOT Z). P(X U Y) = .25 + .36 = 0.61 P(X ∩ Y) = 0 P(X ∩ Y) = .25 x.36 = 0.09 P(X U (Y ∩ Zc) = .25 + .36 x .85 = 0.556

Example 2 : 56% of automobiles in TX are SUVs. 85% of SUV are Black, 2% are White, and 13% are other colors. What is the probability that a randomly chosen automobile will be a SUV that is NOT Black? What is the probability that a randomly chosen automobile is NOT a SUV OR a SUV and its White? What is the probability that when randomly choosing two SUVs, at least one of them is black? P(S ∩ Bc) = .56 x .15 = 0.084 P(SC U (S ∩ W)) = P(SC U (S ∩ W)) = .44 + .56 x .02 = 0.4512 P(At Least 1 Black) = 1 – P(Neither is Black) = 1 – P(Bc ∩ Bc) = 1 – (.15)(.15) = 0.9775

Example 3. The American Red Cross says that about 45% of the US Population has Type O blood, 40% Type A, 11% Type B, and the rest Type AB. a.) Selecting one individual, what is the probability that: 1. has Type AB blood? 2. has Type A or Type B? 3. is NOT Type O? b.) Among four potential donors, what is the probability that: 1. all are Type O? 2. no one is Type AB? 3. at least one person is Type B? P(AB) = 1 – P(OUAUB) = 0.04 P(AUB) = 0.51 P(OC) = 0.55 P(O∩O∩O∩O) = 0.041 P(ABC∩ABC∩ABC∩ABC) = 0.849 1 – P(BC∩BC∩BC∩BC) = 0.373

P(A U B) = P(A)+P(B) – P(A ∩ B) P(Ac ∩ Bc) = 1 – P(A U B) EXAMPLE 2: Let event A = Making a ‘5’ on the AP Stat exam. Let event B = Making a ‘5’ on the AP Calculus exam. If P(A) = 0.14, P(B) = 0.08, and P(A ∩ B) = 0.04 find: 1.) Find the probability that you will make a ’5’ on at least one of the exam. P(A U B)=? 2.) Find the probability that you will NOT make 5 on either exams. P(Ac ∩ Bc) = ? P(A U B) = P(A)+P(B) – P(A ∩ B) 0.18 = 0.14 + 0.08 – 0.04 P(Ac ∩ Bc) = 1 – P(A U B) P(Ac ∩ Bc) = 1 – 0.18 = 0.82

1.) Are events A and B Independent, Mutually Exclusive, or Neither? EXAMPLE 1: If P(A) = 0.40, P(B) = 0.20, and P(A U B) = 0.52 find: P(A ∩ B)=? 1.) Are events A and B Independent, Mutually Exclusive, or Neither? P(A U B) = P(A)+P(B) – P(A ∩ B) 0.52 = 0.40 + 0.20 – P(A ∩ B) P(A ∩ B) = 0.08 ≠ 0 NOT M.E. = 0.08 = P(A)xP(B)

Monty Hall problem Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, which always a goat. He then says to you, "Do you switch your choice or stay?” What should you do? Does it matter?

Monty Hall problem Do NOT Switch Switch Win = 33% Win = 66%

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More with Multiplication Rule 4. A password for a certain computer application MUST be exactly 5 characters long. (Character = Alphabet and Digits) If the first character can not be a number, how many password combinations are possible if: a.) you are NOT allowed to Repeat any character? _____ _____ _____ _____ _____ b.) you are allowed to Repeat? 26 x 35 x 34 x 33 x 32 = 32672640 26 x 36 x 36 x 36 x 36 = 43670016

5. A standard Deck of cards has 52 cards four suits of (2 – 10, J, Q, K, A). Find the Probability of: Selecting ONE card and that card being a: 1.) Heart or a Club 2.) Heart or a Five 3.) Face Card or a Spade 4.) Red Card or a Face Card Selecting TWO cards (With Replacement) and obtaining a: 5.) Heart and then a Face Card 6.) Heart and then a Five 7.) Two Aces 8.) Two Red Cards Selecting TWO cards (Without Replacement) and obtaining a: 9.) Two Aces 10.) Heart and then a Face Card 1/2 4/13 11/26 8/13 3/52 1/52 1/169 1/4 1/221 11/204

Selecting ONE card and that card being a: 1.) Heart or a Club 2.) Heart or a Five 3.) Face Card or a Spade 4.) Red Card or a Face Card Selecting TWO cards (With Replacement) and obtaining a: 5.) Heart and then a Face Card. 6.) Heart and then a Five 7.) Two Aces 8.) Two Red Cards Selecting TWO cards (Without Replacement) and obtaining a: 9.) Two Aces 10.) Heart and then a Face Card