Chapter 13

Towns A, B, C, and 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?

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.

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

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?

7–12 Find the number of distinguishable permutations of the given letters. $$X X Y Y Z Z Z$$

In a family of four children, how many different boy-girl birth-order combinations are possible? (The birth orders BBBG and BBGB are different.)

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

Rolling Dice Six dice are rolled. Find the probability that 2 of them show a four.

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 must pay \(\$ 100\) . What is the expectation of a person playing this game?

A coin is flipped five times, and the resulting sequence of heads and tails is recorded. How many such sequences are possible?

13–18 Evaluate the expression. $$C(8,3)$$

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

Archery An archer hits his target 80\(\%\) of the time. If he shoots 7 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 5 times.

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?

13–18 Evaluate the expression. $$C(9,2)$$

A red die and a white die are rolled, and the numbers showing are recorded. How many different outcomes are possible? (The singular form of the word dice is die.)

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

Television Ratings According to a ratings survey, 40\(\%\) of the households in a certain city tune in to the local evening TV news. If 10 households are visited at random, what is the probability that 4 of them will have their television tuned to the local news?

The registrar at a certain university classifies students according to a major, minor, year (1, 2, 3, 4), and sex (M, F). Each student must choose one major and either one or no minor from the 32 fields taught at this university. How many different student classifications are possible?

Gambling on Stocks An investor buys 1000 shares of a risky stock for \(\$ 5\) a share. She estimates that the probability 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?

A red die and a white die are rolled, and the numbers showing are recorded. How many different outcomes are possible? (The singular form of the word dice is die.)

13–18 Evaluate the expression. $$C(11,4)$$

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. An ace, king, queen, jack, and 10 of the same suit (royal flush)

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?

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?

13–18 Evaluate the expression. $$C(10,5)$$

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?

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.

Blood Type About 45\(\%\) of the population of the United States and Canada have Type O blood. (a) If a random sample of 10 people is selected, what is the probability that exactly 5 have Type \(\mathrm{O}\) blood? (b) What is the probability that at least 3 of the random sample of 10 have Type O blood?

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 this is the only way to win, what is the expected value of this game?

13–18 Evaluate the expression. $$C(100,1)$$

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

A couple intends to have four children. Assume that having a boy or a girl is an equally likely event. (a) List the sample space of this experiment. (b) Find the probability that the couple has only boys. (c) Find the probability that the couple has two boys and two girls. (d) Find the probability that the couple has four children of the same sex. (e) Find the probability that the couple has at least two girls.

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 plays the game many times (in other words, the price at which his expectation is zero).

13–18 Evaluate the expression. $$C(99,3)$$

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

What is the probability that a 13-card bridge hand consists of all cards from the same suit?

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 4 seeds in each hill. (a) What is the probability that least one seed germinates in a given hill? (b) What is the probability that 2 or more seeds will germinate in a given hill? (c) What is the probability that all 4 seeds germinate in a given hill?

19–32 These problems involve permutations. Class Officers In how many different ways can a president, vice president, and secretary be chosen from a class of 15 students?

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.

An American roulette wheel has 38 slots; two slots are numbered 0 and 00, and the remaining slots are numbered from 1 to 36. Find the probability that the ball lands in an odd-numbered slot.

Genders of Children Assume that for any given live hu- man 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 5 children, a majority are boys? (b) What is the probability that in a family of 7 children, a majority are girls?

19–32 These problems involve permutations. Contest Prizes In how many different ways can first, second, and third prizes be awarded in a game with eight contestants?

An all-star baseball team has a roster of seven pitchers and three catchers. How many pitcher-catcher pairs can the manager select from this roster?

A toddler has wooden blocks showing the letters \(C, E, F, H, N,\) and \(R .\) Find the probability that the child arranges the letters in the indicated order. (a) In the order FRENCH (b) In alphabetical order

Genders of Children The ratio of male to female births is in fact not exactly one-to-one. The probability that a newborn turns out to be a male is about \(0.52 .\) A family has 10 children. (a) What is the probability that all 10 children are boys? (b) What is the probability all are girls? (c) What is the probability that 5 are girls and 5 are boys?

19–32 These problems involve permutations. Seating Arrangements In how many different ways can six of ten people be seated in a row of six chairs?

Standard automobile license plates in California display a nonzero digit, followed by three letters, followed by three digits. How many different standard plates are possible in this system?

In the 6/49 lottery game, a player selects six numbers from 1 to 49. What is the probability of picking the six winning numbers?

Education Level In a certain county 20\(\%\) of the population have a college degree. A jury consisting of 12 people is selected at random from this county. (a) What is the probability that exactly 2 of the jurors have a college degree? (b) What is the probability 3 or more of the jurors have a college degree?