Chapter 6 -- Probability Review Questions Addition Rule: or union or & and (in the same problem) P( A B ) = P( A) + P( B) P( A B) *** If the events A and B are mutually exclusive (disjoint), then P ( A B) = 0 *** *** Use Venn Diagrams for these problems*** Mutually Exclusive Not Mutually Exclusive Multiplication Rule: AND intersection P(A B) = P(A)P(B A) P(A B) = P(B)P(A B) When the events A and B are independent, P(B A)=P(B) P(A B)=P(A) Conditional Probability: given A B P(A B) = P( A B) P( B) think of the Venn Diagram ** Note Bayes theorem is a type of conditional probability only use it when the values are unknown for the above formula P(A B)= P(B A)P(A) (tree diagrams are very helpful) P(B A)P(A)+P(A )P(B A )
1.) Find the probability of flipping a head on a coin and rolling a sum of 8 on 2 dice. 2.) A box contains 11 nickels, 4 dimes, and 5 quarters. If you draw 3 coins at random from the box without replacement, what is the probability that you will get a nickel, a dime and a nickel in that order? 3.) A company estimates that 60 % of the adults in the US have seen its TV commercial and that if an adult sees the commercial, there is a 15 % chance that the adult will buy the product. What is the probability that an adult chosen at random in the US will have seen the company s commercial and will have bought its product? 4.) An analysis of the registered voters in the last primary indicated that 55 % of the voters were women. Of the female voters, 35% are registered Democrats, 35 % are registered Republicans, and the rest are assumed independent. Of the male voters, the percentages are 30 %, 45 % and 25 % (D,R,I). Find each probability. (a) a voter chosen at random is a woman (b) A voter chosen at random is a male Republican (c) A Democrat chosen at random is a male 5.) Find the probability of flipping 5 heads on 5 coin tosses. 6.) Find the probability of drawing an ace, then a king out of a standard deck of 52 cards, if the first card is not replaced before the second draw. 7.) 100 teen boys and 100 teen girls were asked if they had ever made a purchase using the Internet. 30 of the boys and 60 of the girls said that they had made purchases. If one of the teens is selected at random (a) What is the probability that they have made a purchase using the Internet? (b) What is the probability that the teen is a girl, given that the person has made a purchase on the Internet? 8.) Donald has ordered a computer and a desk from 2 different stores. Both items are to be delivered on Tuesday. The probability that the computer will be delivered before noon is.6 and the probability that the desk will be delivered before noon is.8. If the probability that the computer or the desk will be delivered before noon is.9, what is the probability that both will be delivered before noon?
9.) When I visit the local library, the probability that someone is reading the current issue of Sports Illustrated is.4, the probability that someone is reading Time is.3, and the probability that at least one of these two magazines is being read by someone is.5. What is the probability that : (a) Both of the magazines are being read? (b) Neither of the two is being read? (c) Exactly one is being read? 10.) Beethoven wrote 9 symphonies and Mozart wrote 27 piano concertos. If a university radio announcer wishes to play first a Beethoven symphony and then a Mozart concerto, in how many ways can this be done? 11.) Television viewers without cable in a certain city can receive 7 different channels. From 5:30 to 6:00 pm on weekdays, 3 channels show news from 6:00-6:30 pm, 5 channels show news; from 6:30 to 7pm, 4 channels show news; and from 7-7:30 pm, 2 channels show news. (a) If a viewing sequence consists of 4 half hour programs between 5:30 and 7:30, how many viewing sequences are there? (b) If a viewing sequence is chosen at random from the set of all possible sequences, what is the probability that all 4 programs selected are news programs? (c) Choosing a sequence as in part (b), what is the probability that a news program is selected at both 6pm and 6:30pm? 12.) Only 1 in 1,000 adults is afflicted with a rare disease for which a diagnostic test has been developed. The test is such that, when an individual actually has the disease, a positive result will occur 99% of the time, while an individual without the disease will show a positive result only 2% of the time. If a randomly chosen individual is tested and the result is positive, what is the probability that the individual has the disease? 13.) A chain of stereo stores is offering a special price on a complete set of components (receiver, compact disc player, speakers, cassette deck.) A purchaser is offered a choice of manufacturer for each component: Receiver: Kenwood, Onkyo, Pioneer, Sony, Sherwood Compact Disc Player: Onkyo, Pioneer, Sony,Technics Speakers: Boston, Infinity, Polk Cassette Deck: Onkyo, Pioneer, Teac, Technics A switchboard display in the store allows a customer to hook together any selection of components (consisting of one of each type) (a) How many different systems are possible? (b) How many different systems are possible if both the receiver and the disc player are to be Sony? (c) How many different systems are possible if none of the components are to be Sony? (d) What is the probability that the system selected contains at least one Sony component?
14.) Among the 400 inmates in the District s jail, some are first offenders, some serve terms of less than five years, and some serve longer terms, with the exact breakdown being: Term < 5 years (S) Longer Terms(L) Totals First Offenders (F) 120 40 Hardened Criminals (H) 80 160 Totals If one of the inmates is selected at random to be interviewed about jail conditions, H is the event that he is a hardened criminal, and L is the event that he is serving a longer term, determine each of the following probabilities directly from the entries and row and column totals of the table: a.) P(H) b.) P(L) c.) P(L and H) d.) P(H and L) e.) P(L H) f.) P(H L) 15.) If P(A)=.8, P(B)=.35, and P(A and B)=.28, check whether events A and B are independent. 16.) If two cards are drawn from an ordinary deck of 52 playing cards, what are the probabilities of getting two diamonds if the drawing is: a.) with replacement b.) without replacement 17.) Brendan has 4 red socks and 6 green socks thrown around in his drawer; the colors are not paired. One dark morning he randomly pulls out 2 socks. What is the probability that he will select a pair of red socks to wear to school? 18.) A telephone company concludes that 70% of its customers pay their monthly bill in full. A more detailed examination of company records indicate that 95% of the customers who pay one monthly bill in full will also pay the next monthly bill in full. Only 10 % of those who pay less than the full amount will pay in full the next month. a. Find the probability that a customer will pay two consecutive months in full. b. Find the probability that the customer will pay neither of the consecutive months in full. c. Find the probability that the customer will pay at least one month in full. d. Find the probability that the customer will pay exactly one month in full.
19.) A game spinner has two equal sections: three numbered 1, one section numbered two and four sections numbered three. The spinner is spun twice. What is the probability that the sum of the two spins is five? 20.) A student placement center has requests from 5 students for interviews regarding employment with a particular consulting firm. Three of these students are math majors and the other two students are statistics majors. Unfortunately, the interviewer has time to talk to only two of the students; these two will be randomly selected from among the five. What is the probability that at least one of the two students selected is a statistics major?