Chapter 14

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. At most two failures

Find the number of distinguishable permutations of the given letters. $$ A A A B B B C C C $$

\(11-12\) . A card is drawn randomly from a standard 52-card deck. Find the probability of the given event. (a) The card drawn is a heart. (b) The card drawn is either a heart or a spade. (c) The card drawn is a heart, a diamond, or a spade.

Lining Up Books In how many ways can five different mathematics books be placed next to each other on a shelf?

Roulette In the game of roulette as played in Las Vegas, the wheel has 38 slots. Two slots are numbered 0 and \(00,\) and the rest are numbered 1 to \(36 .\) A \(\$ 1\) bet on any number other than 0 or 00 wins \(\$ 36\) (S35 plus the S1 bet). Find the expected value of this game.

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. At least two successes

Find the number of distinguishable permutations of the given letters. $$ A A B C D $$

\(13-14\) . A ball is drawn randomly from a jar that contains five red balls, two white balls, and one yellow ball. Find the probability of the given event. (a) A red ball is drawn. (b) The ball drawn is not yellow. (c) A black ball is drawn.

Multiple Routes Towns \(\mathrm{A}, \mathrm{B}, \mathrm{C},\) and \(\mathrm{D}\) are located in such a way that there are four roads from \(A\) to \(B\) , five roads from \(B\) to \(C,\) and six roads from \(C\) to D. How many routes are there from town \(A\) to town D via towns \(B\) and \(C ?\)

Sweepstakes A sweepstakes offers a first prize of \(\$ 1,000,000,\) second prize of \(\$ 100,000,\) and third prize of \(\$ 10,000 .\) Suppose that two million people enter the contest and three names are drawn randomly for the three prizes. (a) Find the expected winnings for a person participating in this contest. (b) Is it worth paying a dollar to enter this sweepstakes?

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. At most three failures

Find the number of distinguishable permutations of the given letters. $$ A B C D D D E E $$

\(13-14\) . A ball is drawn randomly from a jar that contains five red balls, two white balls, and one yellow ball. Find the probability of the given event. (a) Neither a white nor yellow ball is drawn. (b) A red, white, or yellow ball is drawn. (c) The ball that is drawn is not white.

Birth Order In a family of four children, how many different boy-girl birth- order combinations are possible? (The birth orders \(B B B G\) and \(B B G B\) are different.)

A Game of Chance \(A\) box contains 100 envelopes. Ten envelopes contain \(\$ 10\) each, ten contain \(\$ 5\) each, two are "unlucky," and the rest are empty. A player draws an envelope from the box and keeps whatever is in it. If a person draws an unlucky envelope, however, he or she must pay \(\$ 100 .\) What is the expectation of a person playing this game?

Rolling Dice Six dice are rolled. Find the probability that two of them show a four.

Find the number of distinguishable permutations of the given letters. $$ X X Y Y Z Z Z Z $$

A drawer contains an unorganized collection of 18 socks. Three pairs are red, two pairs are white, and four pairs are black. (a) If one sock has been drawn at random from the drawer, what is the probability that is red? (b) Once a sock has been drawn and discovered to be red, what is the probability of drawing another red sock to make a matching pair?

Flipping a Coin \(A\) coin is flipped five times, and the resulting sequence of heads and tails is recorded. How many such sequences are possible?

Combination Lock A safe containing \(\$ 1,000,000\) is locked with a combination lock. You pay \(\$ 1\) for one guess at the six-digit combination. If you open the lock, you get to keep the million dollars. What is your expectation?

Archery An archer hits his target 80\(\%\) of the time. If he shoots seven arrows, what is the probability of each event? (a) He never hits the target. (b) He hits the target each time. (c) He hits the target more than once. (d) He hits the target at least five times.

Find the number of distinguishable permutations of the given letters. $$ X X Y Y Z Z $$

Rolling a Pair of Dice A red die and a white die are rolled, and the numbers that show are recorded. How many different out-comes are possible? (The singular form of the word dice is die.)

Gambling on Stocks An investor buys 1000 shares of a risky stock for \(\$ 5\) a share. She estimates that the probability that the stock will rise in value to \(\$ 20\) a share is 0.1 and the probability that it will fall to \(\$ 1\) a share is \(0.9 .\) If the only criterion for her decision to buy this stock was the expected value of her profit, did she make a wise investment?

Television Ratings According to a ratings survey, 40\(\%\) of the households in a certain city tune in to the local evening TV news. If ten households are visited at random, what is the probability that four of them will have their television tuned to the local news?

Evaluate the expression. $$ C(8,3) $$

A letter is chosen at random from the word EXTRATERRESTRIAL. Find the probability of the given event. (a) The letter \(T\) is chosen. (b) The letter chosen is a vowel. (c) The letter chosen is a consonant.

Rolling Three Dice A red die, a blue die, and a white die are rolled, and the numbers that show are recorded. How many different outcomes are possible?

Slot Machine A slot machine has three wheels, and each wheel has 11 positions: the digits \(0,1,2, \ldots, 9\) and the picture of a watermelon. When a quarter is placed in the machine and the handle is pulled, the three wheels spin independently and come to rest. When three watermelons show, the payout is \(\$ 5 ;\) otherwise, nothing is paid. What is the expected value of this game?

Spread of Disease Health authorities estimate that 10\(\%\) of the raccoons in a certain rural county are carriers of rabies. A dog is bitten by four different raccoons in this county. What is the probability that he was bitten by at least one rabies carrier?

Evaluate the expression. $$ C(9,2) $$

A pair of dice is rolled, and the numbers showing are observed. (a) List the sample space of this experiment. (b) Find the probability of getting a sum of \(7 .\) (c) Find the probability of getting a sum of \(9 .\) (d) Find the probability that the two dice show doubles (the same number). (e) Find the probability that the two dice show different numbers. (f) Find the probability of getting a sum of 9 or higher.

Picking Cards Two cards are chosen in order from a deck. In how many ways can this be done if (a) the first card must be a spade and the second must be a heart? (b) both cards must be spades?

Lottery In a 6\(/ 49\) lottery game, a player pays \(\$ 1\) and selects six numbers from 1 to \(49 .\) Any player who has chosen the six winning numbers wins \(\$ 1,000,000\) . Assuming that this is the only way to win, what is the expected value of this game?

Evaluate the expression. $$ C(11,4) $$

Blood Type About 45\(\%\) of the population of the United States and Canada have Type \(\mathrm{O}\) blood. (a) If a random sample of ten people is selected, what is the probability that exactly five have Type O blood? (b) What is the probability that at least three of the random sample of ten have Type O blood?

\(19-22\) . A poker hand, consisting of five cards, is dealt from a standard deck of 52 cards. Find the probability that the hand contains the cards described. Five hearts

Choosing Outfits A girl has 5 skirts, 8 blouses, and 12 pairs of shoes. How many different skirt-blouse-shoes outfits can she wear? (Assume that each item matches all the others, so she is willing to wear any combination.)

A Game of Chance A bag contains two silver dollars and six slugs. A game consists of reaching into the bag and drawing a coin, which you get to keep. Determine the "fair price" of playing this game, that is, the price at which the player can be expected to break even if he or she plays the game many times (in other words, the price at which the player's expectation is zero).

Evaluate the expression. $$ C(10,5) $$

Handedness A psychologist needs 12 left-handed subjects for an experiment, and she interviews 15 potential subjects. About 10\(\%\) of the population is left- handed. (a) What is the probability that exactly 12 of the potential subjects are left-handed? (b) What is the probability that 12 or more are left-handed?

\(19-22\) . A poker hand, consisting of five cards, is dealt from a standard deck of 52 cards. Find the probability that the hand contains the cards described. Five cards of the same suit

ID Numbers A company's employee ID number system consists of one letter followed by three digits. How many different ID numbers are possible with this system?

Evaluate the expression. $$ C(100,1) $$

Germination Rates \(\quad\) A certain brand of tomato seeds has a 0.75 probability of germinating. To increase the chance that at least one tomato plant per seed hill germinates, a gardener plants four seeds in each hill. (a) What is the probability that at least one seed will germinate in a given hill? (b) What is the probability that two or more seeds will germinate in a given hill? (c) What is the probability that all four seeds will germinate in a given hill?

\(19-22\) . A poker hand, consisting of five cards, is dealt from a standard deck of 52 cards. Find the probability that the hand contains the cards described. Five face cards

ID Numbers A company has 2844 employees. Each employee is to be given an ID number that consists of one letter followed by two digits. Is it possible to give each employee a different ID number using this scheme? Explain.

Lightning Insurance An insurance company has determined that in a certain region the probability of lightning striking a house in a given year is about \(0.0003,\) and the aver- age cost of repairs of lightning damage is \(\$ 7500\) per incident. The company charges \(\$ 25\) per year for lightning insurance. (a) What is the company's expected value for the net income from each lightning insurance policy? (b) If the company has \(450,000\) lightning damage policies, what is the company's expected yearly income from lightning insurance?

Evaluate the expression. $$ C(99,3) $$

Genders of Children Assume that for any given live human birth, the chances that the child is a boy or a girl are equally likely. (a) What is the probability that in a family of five children a majority are boys? (b) What is the probability that in a family of seven children a majority are girls?