Probability II

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}

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 an even # EC={1,3,5}

Union the event A or B happening consists of all outcomes that are in at least one of the two events Rolling a prime # or even number E={2,3,4,5,6}

Intersection the event A and B happening consists of all outcomes that are in both events Drawing a red card and a “2” E = {2 hearts, 2 diamonds}

Mutually Exclusive (disjoint) two events have no outcomes in common Roll a “2” and a “5”

Probability Denoted by P(Event) This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely.

Experimental Probability The relative frequency at which a chance experiment occurs Flip a fair coin 30 times & get 17 heads

Law of Large Numbers As the number of repetitions of a chance experiment increase, the difference between the relative frequency of occurrence for an event and the true probability approaches zero.

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 Complement of an event….. Ec

Independent So Chuck, what are you going to do next year after graduation? Oh, remember – I’m going to college to become an engineer. Oh, I plan on going to a culinary school to become a chef – I just love to cook! Are Chuck’s responses independent? Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs Teacher to student in school Teacher to student in front of parent

Rule 4. Multiplication If two events A & B are independent, General rule:

What does this mean? Independent? Yes

Ex. 1) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. What is the probability that both bulbs are defective? Can you assume they are independent?

Independent? Yes No

Ex. 2) There are seven girls and eight boys in a math class Ex. 2) There are seven girls and eight boys in a math class. The teacher selects two students at random to answer questions on the board. What is the probability that both students are girls? Are these events independent?

P(Spade & Spade) = 1/4 · 12/51 = 1/17 Ex 6) 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) P(Spade & Spade) = 1/4 · 12/51 = 1/17 The probability of getting a spade given that a spade has already been drawn.

Rule 5. Addition 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)

What does this mean? Mutually exclusive? Yes

Are these disjoint events? Ex 3) A large auto center sells cars made by many different manufacturers. Three of these are Honda, Nissan, and Toyota. (Note: these are not simple events since there are many types of each brand.) 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

Mutually exclusive? Yes No

Ex. 4) Musical styles other than rock and pop are becoming more popular. A survey of college students finds that the probability they like country music is .40. The probability that they liked jazz is .30 and that they liked both is .10. What is the probability that they like country or jazz? P(C or J) = .4 + .3 -.1 = .6

Mutually exclusive? Yes No Independent? Yes

Disjoint events are dependent! Ex 5) If P(A) = 0.45, P(B) = 0.35, and A & B are independent, find P(A or B). Is A & B disjoint? NO, independent events cannot be disjoint If A & B are disjoint, are they independent? Disjoint events do not happen at the same time. So, if A occurs, can B occur? Disjoint events are dependent! P(A or B) = P(A) + P(B) – P(A & B) If independent, multiply How can you find the probability of A & B? P(A or B) = .45 + .35 - .45(.35) = 0.6425

Flipping a coin and rolling a die are independent events. If a coin is flipped & a die rolled at the same time, what is the probability that you will get a tail or a number less than 3? Sample Space H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6 T1 T2 T3 T4 T5 T6 H1 H3 H2 H5 H4 H6 Coin is TAILS Roll is LESS THAN 3 Flipping a coin and rolling a die are independent events. Independence also implies the events are NOT disjoint (hence the overlap). Count T1 and T2 only once! P (tails or even) = P(tails) + P(less than 3) – P(tails & less than 3) = 1/2 + 1/3 – 1/6 = 2/3

Ex. 7) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. What is the probability that exactly one bulb is defective? P(exactly one) = P(D & DC) or P(DC & D) = (.05)(.95) + (.95)(.05) = .095

Ex. 8) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. What is the probability that at least one bulb is defective? P(at least one) = P(D & DC) or P(DC & D) or (D & D) = (.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 one bulb is defective? Ex. 8 revisited) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. What is the probability that at least one bulb is defective? P(at least one) = 1 - P(DC & DC) .0975

Ex 9) 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? Dr. Pepper P(at least one winning symbol) = 1 – P(no winning symbols) 1 - .96 = .4686

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

Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is .51 and is a girl and plays sports is .10. If the student is female, what is the probability that she plays sports?

Ex 11) The probability that a randomly selected student plays sports if they are male is .31. What is the probability that the student is male and plays sports if the probability that they are male is .49?