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

If AP Statistics Review sessions were offered, would you likely attend a: A Saturday Morning Session 9 am to Noon B After School Session 4:20 – 6:20 C I Would probably NOT attend either.

If you answered “B” (Weekday afternoon 4:20-6:20), which day of the week best fits you schedule. A Monday B Tuesday C Wednesday D Thursday

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 ∩ B) = 0.80 1.) Are events A and B Independent, Mutually Exclusive, or Neither ? If Mutually Exclusive: P(A ∩ B) = 0 If Independent: P(A ∩ B) = P(A) ∙ P(B)

a.) If X and Y are Mutually Exclusive Find the Probability of X or Y EXAMPLE 2: 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 3 : 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

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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%

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