1. 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 – ( ) = 0.76
= 0.96
(.04)(.04) =
No Bugs on TWO disks = (.76)(.76)=.5776
P(At least One)= 1 – =

3. A B C D VENN DIAGRAM 55 – 12 = 43 125 – 12 = 113 = 12
1600 – ( ) = 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

4. 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?

5. a.) If X and Y are Mutually Exclusive Find the Probability of X or Y
EXAMPLE 1: Given that P(X) = 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) = = 0.61
P(X ∩ Y) = 0
P(X ∩ Y) = .25 x.36 = 0.09
P(X U (Y ∩ Zc) = x .85 = 0.556

6. 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)) = x .02 =
P(At Least 1 Black) = 1 – P(Neither is Black) =
1 – P(Bc ∩ Bc) = 1 – (.15)(.15) =

7. 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

8. 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) = 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)
= – 0.04
P(Ac ∩ Bc) = 1 – P(A U B)
P(Ac ∩ Bc) = 1 – 0.18 = 0.82

9. 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) = 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)
= – P(A ∩ B)
P(A ∩ B) = 0.08 ≠ 0 NOT M.E.
= 0.08 = P(A)xP(B)

10. 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?

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

12. Page 341
20, 21, 23-25, 27, 28, 31, 33, 34

14. 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 x x x =
26 x x x x =

15. 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

16. 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