Stats #3 quiz chap 8 and 9

Rules of Probabilities

1. The probability of any event E, P(E), must be greater than or equal to 0 and less than or equal to 1 2. The sum of the probabilities of all outcomes must equal 1

how many cards in deck and suits

52 cards and four sets (two back and two red)

A recent Harris Poll survey of 1010 U.S. adults selected at random showed that 627 consider the occupation of firefighter to have very great prestige. Estimate the probability that a U.S. adult selected at random thinks the occupation of firefighter has very great prestige. Is this empirical or classical?

627/1010 this is empirical

Probability of "at least one": HH, HT, TH, TT P(at least one tail)

=1-P(No Tails) P(at least one)= 1-p(none)

The Law of Large Numbers

As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome

Are the following events independent or dependent? • Drawing a 5 card poker hand • Rolling two fair dice (or rolling one die twice) • Choosing 5 students from a class for a committee

-Dependent -Independent -Dependent

Do the following represent probability models? Colors in bag of M and Ms Brown 0.12 Yellow 0.15 Red 0.12 Blue 0.23 Orange 0.23 Green 0.15 Survey on Seat Belt Use while Driving Never 0.02 Rarely 0.05 Sometimes 0.08 Most of time 0.15 Always 0.68

-If adds up to 1 then its probability model -colors in bag of M and Ms --> yes bc =1 -survey seat belt use while driving> =.98 so no

Sampling With Replacement 4 cards are drawn from a standard deck of 52 cards. Each time a card is drawn you make a note as to what the card is, put it back in the deck, reshuffle, and draw the next card. (This is drawing with replacement.) • What is the probability all 4 cards are hearts? • What is the probability all 4 cards are Aces?

-Independent -P(H1 and H2 and H3 and H4) = 13/52 x 13/52 x 13/52 x 13/52 = 1/4 = 1/256 = .00391

Probability Model

-List all outcomes/probability (Rules most hold) with corresponding probabilities -If adds up to 1 then its probability model -Are all numbers between 0 and 1?

Sampling Without Replacement 4 cards are drawn from a standard deck of 52 cards. Each time you draw, you keep the card and then draw the next card. (This is sampling without replacement). • What is the probability all 4 cards are hearts? • What is the probability all 4 cards are Aces? • What is the probability none of the 4 cards are Aces?

-P(H1 and H2 and H3 and H4) 13/52 x 12/51 x 11/50 x 10/49 = .00264 -P(A1 and A2 and A3 and A4) 4/52 x 3/51 x 2/50 x 1/49

A company is testing a new medicine for migraine headaches. In the study, 150 women were given the new medicine and an additional 100 women were given a placebo. Each participant was directed to take the medicine when the first symptoms of a migraine occurred and to record whether the headache when away or lingered. Headache went away Headache lingered Headache Went Away Lingered Given medicine 132 18 Given placebo 56 44 If a study participant is selected at random, • What is the probability she was given a placebo? • What is the probability her headache went away within 45 minutes? • What is the probability she was given a placebo and her headache went away? • What is the probability she was given the placebo or her headache went way in 45 minutes?

-P(Placebo)= 100/250 = 10/25 = 2/5 -P(went away)= 132+56/250 = 188/250 = 94/125 -P(Placebo and went away)= 56/250 = 28/125 -P(Placebo or away)= P(placebo) + P(away) - P(P(Placebo and away)) = 100/250 + 188/250 - 56/250= 232/250= 110/125

The question "Do you smoke?" was asked of 100 people. Yes No Male 19 41 Female 12 28 What is the probability that • a randomly selected individual smokes? • a randomly selected individual is male? • a randomly selected individual is male, given the individual smokes? • a randomly selected male smokes?

-P(Smokes)= 19+12/100= 31/100 -P(male)= 19+41/100= 60/100 -P(male/smoke)= 19/19+12=19/31 -P(smoke/male)= 19/19+12

According to the U. S. bureau, the probability that a randomly selected household speaks only English at home is 0.81. The probability that a randomly selected household speaks only Spanish at home is 0.12. What is the probability that a randomly selected household speaks • only English or only Spanish at home? • a language other than only English or only Spanish at home? • a language other than only English at home? • Can the probability that a randomly selected household speaks only Polish at home equal 0.08? Why or why not

-P(only English or only Spanish) = .81+.12=.93 -1-.93=.07 -1-.81=.19 -No, P(only Polish) < .07

Classical Method of Probability

-based on equally likely outcomes - # outcomes that lead success/ total # of outcomes

Empirical Approach to Probability

-based on probability - # successes/ total # of trials

Are the following events mutually exclusive? • Drawing a single card from a standard deck and drawing a King or drawing an Ace. • Drawing a single card from a standard deck and drawing a King or drawing a heart. • Rolling two dice and rolling a nine or rolling a double (both dice show the same number).

-mutually exclusive -not mutually exclusive -mutually exclusive

The General Addition Rule

For any two events E and F, P(E or F)= P(E)+ P(F)− P(E and F)

Conditional Probability Rule

If E and F are any two events, then P(F/E)= P(E and F)/P(E) = N(E and F)/ N(E) The probability of event F occurring, given the occurrence of event E, is found by dividing the probability of E and F by the probability of E, or by dividing the number of outcomes in E and F by the number of outcomes in E

Multiplication Rule for n Independent Events

If E1, E2, E3, ... and En are independent events, then P( E1 and E2 and E3 and ... and En ) = P(E1 )x P(E2 ).....P(En )

Complement of an event

Let S denote the sample space of a probability experiment and let E denote an event. The complement of E, denoted E C , is all outcomes in the sample space S that are not outcomes in the event E.

A veterinarian tells you that if you breed two cream-colored guinea pigs, the probability that an off-spring will be pure white is 0.25. Assume each birth by a pair guinea pigs is independent. • If your guinea pigs have 3 offspring, what is the probability they will all be white? • What is the probability an offspring will not be pure white? • What is the probability none of the 3 offspring will not be pure white? • If your guinea pigs have 3 offspring, what is the probability that at least one will be pure white?

P(W)=.25=1/4 -P(W1 and W2 and W3)= 1/4 x 1/4 x 1/4 = 1/64 -P(not W)= 1-1/4= 4/4-1.4=3/4 -P(not W1 and not W2 and not W3)= 3/4x3/4x3/4=27/64 -1, 2, or 3 pare white P(at least one W)= 1-P(none white)= 1-27/64=64/64-27/64= 37/64

A fair coin is tossed 3 times. a. List the sample space. b. What is the probability of exactly 2 heads? Is this empirical or classical?

a. {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} b. 3/8 c. classical

P(E of F) think

addition

Two events are blank if the occurrence of event E in a probability experiment affects the probability of event F.

dependent

Two events are blank if they have no outcomes in common.

disjoint

Two events E and F are blank if the occurrence of event E in a probability experiment does not affect the probability of event F.

independent

Subjective Probability

just what we think

A card is drawn from a standard deck. Find the probability of drawing a King or drawing an Ace.

mutually exclusive 4/52 + 4/52 (P(king)+P(Ace))= 8/52

Another name for disjoint events is

mutually exclusive events

A card is drawn from a standard deck. What is the probability the card was a King or a heart.

not mutually exclusive P(king or heart)= 4/52 + 13/52 - 1/52

probability of certain event

one

A game is played by flipping a coin and then rolling a die. List the sample space.

sample space {H1, H2 H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

Probability

the likelihood a given event will happen

unusual event

when probability of event less than .05

Probability of impossible events

zero