Probability I

Notes Probability of A occurring P(A) Sum of all possible outcomes = 1

Sample Space Roll a die S={1,2,3,4,5,6} the collection of all possible outcomes of a chance experiment Roll a die S={1,2,3,4,5,6}

# Of Occurrences of Event Relative Frequency # Of Occurrences of Event #Trials Not rolling a even # EC={1,3,5}

The Law of Large Numbers The long run relative frequency will approach the actual probability as the number of trails increases Coins? 2, 10, 20.

Event any collection of outcomes from the sample space Rolling a prime # E= {2,3,5}

Complement Consists of all outcomes that are not in the event Not rolling a even # EC={1,3,5} P(A) = 1 – P(A)

Mutually Exclusive (disjoint) two events have no outcomes in common Roll a “2” or a “5” Draw a Black card or a Diamond

Not -Mutually Exclusive (Non- disjoint) two events have outcomes in common Draw a Black card or a Spade

Union—Disjoint the event A or B happening consists of all outcomes that are in at least one of the two events Draw a Black card or a Diamond

Union—Disjoint Draw a Black card or a Diamond P(B U D) = P(B) + P(D)

Intersection the event A and B happening consists of all outcomes that are in both events Draw a Black card and a 7

Intersection P(B S) = P(B)•P(S) Draw a Black card and a 7 U

Union—Not Disjoint the event A or B happening BUT WE CAN’T Double Count! Draw a Black card or a 7 P(B or 7) = P(B) + P(7) – P(B and 7)

Venn Diagrams Used to display relationships between events Helpful in calculating probabilities

Venn Diagram Mutually Exclusive / Disjoint events A B

Venn Diagram Not Mutually Exclusive / Non- Disjoint events A B

Venn diagram - Complement of A

Venn diagram - A and B A B

Statistics & Computer Science & not Calculus Com Sci Statistics & Computer Science & not Calculus

(Statistics or Computer Science) and not Calculus Com Sci Com Sci (Statistics or Computer Science) and not Calculus

Two- Way Table P ( has pierced ears. ) P( is a male or has pierced ears. ) P( is a female or has pierced ears )

Basic Rules of Probability Rule 1. Legitimate Values For any event E, 0 < P(E) < 1 Rule 2. Sample space If S is the sample space, P(S) = 1

Rule 3. Complement For any event E, P(E) + P(not E) = 1 Or P(not E) = 1 – P(E)

Rule 4. Addition (A or B) If two events E & F are disjoint, P(E or F) = P(E) + P(F) (General) If two events E & F are not disjoint, P(E or F) = P(E) + P(F) – P(E & F)

Are these disjoint events? Ex 1) A large auto center sells cars made by many different manufacturers. Three of these are Honda, Nissan, and Toyota. Suppose that P(H) = .25, P(N) = .18, P(T) = .14. Are these disjoint events? yes P(H or N or T) = .25 + .18+ .14 = .57 P(not (H or N or T) = 1 - .57 = .43

Independent Independent Not independent Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs Flip a Coin and Get Heads. Flip a coin again. P(T) Draw a 7 from a deck. Draw another card. P(8) Independent Not independent

Rule 5. Multiplication If two events A & B are independent, General rule:

The probability that a student will receive a state grant is 1/3, while the probability she will be awarded a federal grant is ½. If whether or not she receives one grant is not influenced by whether or not she receives the other, what is the probability of her receiving both grants?

Suppose a reputed psychic in an extrasensory perception (ESP) experiment has called heads or tails correctly on TEN successive coin flips. What is the probability that her guessing would have yielded this perfect score?

Tree Diagrams Consider flipping a coin twice. What is the probability of getting two heads? Sample Space: HH HT TH TT

Tree Diagrams Getting Tails Twice

Example: Teens with Online Profiles The Pew Internet and American Life Project finds that 93% of teenagers (ages 12 to 17) use the Internet, and that 55% of online teens have posted a profile on a social-networking site. What percent of teens are online and have posted a profile? 51.15% of teens are online and have posted a profile.

Ex. 3) A certain brand of cookies are stale 5% of the time Ex. 3) A certain brand of cookies are stale 5% of the time. You randomly pick a package of two such cookies off the shelf of a store. What is the probability that both cookies are stale? Can you assume they are independent?

Read “probability of B given that A occurs” Ex 5) Suppose I will pick two cards from a standard deck without replacement. What is the probability that I select two spades? Are the cards independent? NO P(A & B) = P(A) · P(B|A) Read “probability of B given that A occurs” P(Spade & Spade) = 1/4 · 12/51 = 1/17 The probability of getting a spade given that a spade has already been drawn.

P(exactly one) = P(S & SC) or P(SC & S) = (.05)(.95) + (.95)(.05) Ex. 6) A certain brand of cookies are stale 5% of the time. You randomly pick a package of two such cookies off the shelf of a store. What is the probability that exactly one cookie is stale? P(exactly one) = P(S & SC) or P(SC & S) = (.05)(.95) + (.95)(.05) = .095

Ex. 7) A certain brand of cookies are stale 5% of the time Ex. 7) A certain brand of cookies are stale 5% of the time. You randomly pick a package of two such cookies off the shelf of a store. What is the probability that at least one cookie is stale? P(at least one) = P(S & SC) or P(SC & S) or (S & S) = (.05)(.95) + (.95)(.05) + (.05)(.05) = .0975

Rule 6. At least one The probability that at least one outcome happens is 1 minus the probability that no outcomes happen. P(at least 1) = 1 – P(none)

What is the probability that at least cookie is stale? Ex. 7 revisited) A certain brand of cookies are stale 5% of the time. You randomly pick a package of two such cookies off the shelf of a store. What is the probability that at least cookie is stale? P(at least one) = 1 – P(SC & SC) .0975

1 - .96 = .4686 P(at least one winning symbol) = Ex 8) For a sales promotion the manufacturer places winning symbols under the caps of 10% of all Dr. Pepper bottles. You buy a six-pack. What is the probability that you win something? P(at least one winning symbol) = 1 – P(no winning symbols) Dr. Pepper 1 - .96 = .4686

Warm Up Allergies Female Male Total 10 8 18 13 9 22 23 17 40 Allergies No Allergies 13 9 22 23 17 40 What is the probability of not having allergies? What is the probability of having allergies if you are a male? Are the events “Female” and “allergies” independent? Justify your answer.

Conditional Probability and Independence Handedness Female Male Total Left 3 1 __ Right 18 8 Are the events “female” and “right handed” independent?

Rule 7: Conditional Probability A probability that takes into account a given condition

. What is the probability that a randomly selected resident who reads USA Today also reads the New York Times? There is a 12.5% chance that a randomly selected resident who reads USA Today also reads the New York Times.

Using Table D… When performing a random simulation we can use Table D. Lets say I have a 30% Chance of winning a class lottery.

Probabilities from two way tables Stu Staff Total American 107 105 212 European 33 12 45 Asian 55 47 102 Total 195 164 359 What is the probability that the driver is a student?

Probabilities from two way tables Stu Staff Total American 107 105 212 European 33 12 45 Asian 55 47 102 Total 195 164 359 What is the probability that the driver is staff and drives an Asian car?

Probabilities from two way tables Stu Staff Total American 107 105 212 European 33 12 45 Asian 55 47 102 Total 195 164 359 If the driver is a student, what is the probability that they drive an American car? Condition

Whiteboard Challenge

The probability of any outcome of a random phenomenon is (a) the precise degree of randomness present in the phenomenon. (b) any number as long as it is greater than 0 and less than 1. (c) either 0 or 1, depending on whether or not the phenomenon can actually occur or not. (d) the proportion of times the outcome occurs in a very long series of repetitions. (e) none of the above.

A randomly selected student is asked to respond Yes, No, or Maybe to the question “Do you intend to vote in the next presidential election?” The sample space is { Yes, No, Maybe }. Which of the following represents a legitimate assignment of probabilities for this sample space? 0.4, 0.4, 0.2 0.4, 0.6, 0.4 0.3, 0.3, 0.3 0.5, 0.3, –0.2 1⁄4, 1⁄4, 1⁄4

You play tennis regularly with a friend, and from past experience, you believe that the outcome of each match is independent. For any given match you have a probability of 0.6 of winning. The probability that you win the next two matches is (a) 0.16. (b) 0.36. (c) 0.4. (d) 0.6. (e) 1.2.

There are 10 red marbles and 8 green marbles in a jar There are 10 red marbles and 8 green marbles in a jar. If you take three marbles from the jar (without replacement), the probability that they are all red is: (a) 0.069 (b) 0.088 (c) 0.147 (d) 0.171 (e) 0.444

Jolor and Mi Sun are applying for summer jobs at a local restaurant Jolor and Mi Sun are applying for summer jobs at a local restaurant. After interviewing them, the restaurant owner says, “The probability that I hire Jolor is 0.7, and the probability that I hire Mi Sun is 0.4. The probability that I hire at least one of you is 0.9.” What is the probability that both Jolor and Mi Sun get hired? (a) 0.1 (b) 0.2 (c) 0.28 (d) 0.3 (e) 1.1

Select a random integer from –100 to 100 Select a random integer from –100 to 100. Which of the following pairs of events are mutually exclusive (disjoint)? (a) A: the number is odd; B: the number is 5 (b) A: the number is even; B: the number is greater than 10 (c) A: the number is less than 5; B: the number is negative. (d) A: the number is above 50; B: the number is less than 20. (e) A: the number is positive; B: the number is odd.

A recent survey asked 100 randomly selected adult Americans if they thought that women should be allowed to go into combat situations. Here are the results, classified by the gender of the subject: Gender Yes No Male 32 18 Female 8 42 The probability of a “Yes” answer, given that the person was Female, is (a) 0.08 (b) 0.16 (c) 0.20 (d) 0.40 (e) 0.42

A recent survey asked 100 randomly selected adult Americans if they thought that women should be allowed to go into combat situations. Here are the results, classified by the gender of the subject: Gender Yes No Male 32 18 Female 8 42 ______________________________________________ The probability that a randomly selected subject in the study is Male or answered “No” is: (a) 0.18 (b) 0.36 (c) 0.68 (d) 0.92 (e) 1.10

An airline estimates that the probability that a random call to their reservation phone line result in a reservation being made is 0.31. This can be expressed as P(call results in reservation) = 0.31. Assume each call is independent of other calls. Describe what the Law of Large Numbers says in the context of this probability.

An airline estimates that the probability that a random call to their reservation phone line result in a reservation being made is 0.31. This can be expressed as P(call results in reservation) = 0.31. Assume each call is independent of other calls. What is the probability that none of the next four calls results in a reservation?

An airline estimates that the probability that a random call to their reservation phone line result in a reservation being made is 0.31. This can be expressed as P(call results in reservation) = 0.31. Assume each call is independent of other calls. You want to estimate the probability that exactly one of the next four calls result in a reservation being made. Describe the design of a simulation to estimate this probability. Explain clearly how you will use the partial table of random digits below to carry out five simulations. 188 87370 88099 89695 87633 76987 85503 26257 51736 189 88296 95670 74932 65317 93848 43988 47597 83044 190 79485 92200 99401 54473 190 34336 82786 05457 60343 191 40830 24979 23333 37619 56227 95941 59494 86539 192 32006 76302 81221 00693 95197 75044 46596 11628