Preparation W09 Math 108 Permutations

b. If patrons in a casino spin the wheel 100,000 times, how many times should you expect a red number?

he law of large numbers tells us that as the game is played more and more times, the proportion of times that the wheel shows a red number should get closer to 0.474. In 100,000 tries, the wheel should come up red close to 47.4% of the time, or about 47,400 times.

Compute the following without using the factorial function on your calculator: 10!/2! (10-2)!

10x9x8x7x6x5x4x3x2x1/ 8x7x6x5x4x3x2x1, 2x1 8-1 cancel on top and bottom leaving 10x9/2x1 = 45

Little League manager has 15 children on her team. How many ways can she form a 9-player batting order?

15P9 = 15!/6! = 1,816,214,400

The probability that some pair of students in the class of 25 shares the same birthday?

The pair sharing the birthday does not have to include you. But, it is still an at least once question, because we are looking for the probability of a shared birthday among at least one pair of students. P(at least one pair of shared birthdays) = 1 - P(no shared birthdays).

Suppose you flip a coin 10,000 times. Considering the Law of Large Numbers, select the most appropriate statement below:

The ratio of heads to the total flips should get closer to 1/2 as you continue to flip the coin.

Suppose you flip a fair coin. What is the probability of getting heads?

.50

Five members of a city coucil decide they need a three-person committe to study the impact of a new shopping center. Ursula, Vern, Wendy, Yolanda, and Zeke. How many committees could be formed from the five council member?

First, find the number of permutations of the n= 5 council members selected r = 3 at a time: 5P3 = 5!/ (5-3)! = 5!/2! = 5 x 4 x 3 = 60 There are 60 possible permutatios. However, order matters for permutations but not for committees. So, any three-person committee can be listed in 3! = 3x2x1 = 6 different orders. Because each three-person committee is counted 3! = 6 times by the permutations formula, this formula gives us six times the actual number of committees. We must, therefore, divide the number of permutations by 3! to find the number of committees: 60/ 3 x 2 x 1 = 10

Imagine you are doing research for a term paper in your English class and you have chosen the topic to be "The Benefits and Costs of My State's Lottery". You find the following information for your state's lottery: Lottery summary Prize Probability Jackpot 1 in 200,000,000 $11,000 1 in 180,000 $700 1 in 12,000 $7 1 in 170 $3 1 in 50 You know that it costs $3 to purchase the lottery ticket in order to play the game. What would you conclude in your term paper regarding the expected value of the game (including purchasing the ticket), if the jackpot were to be set at 180,000,000?

-3 x 1 = -3 3 x 1/50 = .06 7x 1/170 = .041176471 700 x 1/12000 = .058333333 11000 x 1/180000 = .061111111 180,000,000 x 1/200,000,000 = .9 Add them all up = -1.879 or -1.88

Your friend has 20 different magazines and 6 different DVDs he is willing to loan you. Compute the following: 1. If you take one magazine and one DVD, how many different collections could you end up with? 2. If you take 5 magazines, how many collections of magazines are possible? 3. If you take 4 magazines and you are particular about what order you read them in, how many possible orders are there? 4. If you take 3 magazines and 3 DVDs, how many different collections could you end up with?

1. 20 magazines x 6 DVDs = 120 combinations 2. 20 magazines taken 4 at a time: 20!/(20-5)! x 5! = 15,504 3. 20 magazines taken 4 at a time and order matters: 20!/(20-4)! = 116,280 4. 20!/(20-3)! x 3! = 1140 6!/(6-3)! x 3! = 20 multiply both answers = 22,800 different collection

If you toss a coin 9 times and list the results (heads or tails) in order, how many different orders are possible?

2 choices and 9 tosses = 2^9 = 512

You are playing a board game where you roll a pair of dice about 250 times. During the course of the game, you roll doubles two times in a row. What is the probability of rolling doubles two times in a row? Should you be surprised that this happened in 250 rolls?

2/12 reduces to 1/6 1/6 x 1/6 = .0278 You should not be surprised since streaks like this are likely to happen.

A manufacturer's serial numbers consist of 4 letters of the alphabet followed by 2 numerical digits. How many serial numbers are possible?

26 letters in 4 spots and 10 numbers in two spots 26x26x26x26 x 10x10=45,697,600

How many different arrangements could be made using 3 characters if repetition is not allowed? Assume your "alphabet" consists of the lowercase letters.

26 letters selected 3 at a time 26!/26-3! = 26!/23! = 15,600

How many six-character passwords can be made by combining lowercase letters, uppercase letters, numerals, and the characters @, $, and &?

26 lowercase letters, 26 upercase letters, 10 numerals, and three symbols, making atotal of 65 characters to choose from. For six-character passwords, we select r = 6 symbols with n = 65 choices for each character. n^r = 65^6 = 75,418,890,625

Suppose Fred randomly thinks of 5 different whole numbers between 1 and 33 inclusive. What is the probability that George can correctly guess all 5 numbers?

33C5 = 33!/(33-5)! x 5! = 237,336 There is only one correct way to guess exactly all 5 numbers. Expressed as a fraction 1/237,336

In how many ways can 7 graduates line up to receive their diplomas if there are 4 girls and 3 boys and the girls will receive their diplomas first?

4!3! There are 4 ways the first girl can be chosen. 3 ways the 2nd one can be chosen 2 ways the third girl can be chosen 1 way that the last girl can be chosen 3 ways the first boy can be chosen etc 4 x 3 x2x 1x 3x2x1 = 144

If there are 40 runners in a race and prizes will be awarded to first, second, and third places only, what is the probability of correctly predicting the winners in order?

40 runners selected 3 at a time order important 40P3 40!/(40-3)! =59,280 Only one way to predict winning order expressed as 1/59,280

Compute the following without using the factorial function on your calculator: 7!/5!

42

In how many ways can 7 graduates line up to receive their diplomas if there are 4 girls and 3 boys and the girls will receive their diplomas first?

4x3x2x1 x 3x2x1 = 144 ways

Compute the following without using the factorial function on your calculator: 6! =

6x5x4x3x2x1= 720

A mother chooses 3 of her 7 children to help cook dinner. How many different groups of children can she choose?

7 children with 3 chosen at a time = 7P3 7!/(7-3)! x 3! = 35

In how many ways can 7 graduates line up to receive their diplomas?

7x6x5x4x3x2x1 = 5040 or 7!

A soccer coach who has 15 children on her team will be playing 7 children at a time, each at a distinct position. Which number is largest? A) The number of permutations of the 7 positions that are possible with the 15 children. B) The number of different ways of arranging the 7 children playing at any one time among the 7 positions. C) The number of combinations of 7 children that can be chosen from the 15.

A) The number of permutations of the 7 positions that are possible with the 15 children. 15!/(15-7)! - 15!/8! = 32,432,400 B) The number of different ways of arranging the 7 children playing at any one time among the 7 positions. C) The number of combinations of 7 children that can be chosen from the 15. 15!/(8)! x (7)! = 6435

You are betting on a game in which each bet has an expected value of -$0.33. This means that A) if you play the game many times, on average you will have lost about 33 cents per game. B) you will win 33 cents every time you play. C) you will lose 33 cents every time you play.

A) if you play the game many times, on average you will have lost about 33 cents per game.

One person in a stadium filled with 100,000 people is chosen at random to win a free pair of airline tickets. The probability that someone will win the tickets A) is 1. B) is 1 in 100,000. C) depends on the cost of the plane tickets.

A) is 1. Consider the probability of the person who wins the airline tickets is 1 in 100,000. But the probability that someone will win is 1 since this is a sure event. Given that the lucky draw is conducted, someone will surely be the winner.

Cameron is betting on a game in which the probability of winning is 1 in 5. He's won five games in a row so he decides to double his bet on the sixth game. This strategy shows A) poor logic, as he has an 80% chance of losing the double bet. B) poor logic, because on such a good day he should bet much more. C) good logic, since he's having a good day and will probably win again.

A) poor logic, as he has an 80% chance of losing the double bet. Cameron has been winning in a betting game with the probability of winning of 1/5 for 5 games. Doubling his bets on the 6th game believing that he's having a winning streak is a wrong move as the probability of winning the 6th game is the same that is 1/5 or 20% and his chances of losing the double bet are still high that is 80% .

A waitress has three different entrees for the three people sitting at a table, but has forgotten which person ordered which entree. How many possible ways are there for her to serve the entrees? A) 6 B) 3 C) 33

A) 6 3x2x1

A middle school principal needs to schedule six different classes - algebra, English, history, Spanish, science, and gym - in six different time periods. How many different class schedules are possible?

All selections come from the same group of classes, each class is scheduled only once, and the order of the classes matter (because a schedule that begins with Spanish is different from one that begins with gym). The principal may choose any of the six classes for the first period, leaving five choices left for the second period. And, three choices for the fourth period, two choices for the fifth period, and only one choice for the sixth period. 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 different ways.

Consider a lottery with 100 million tickets in which each ticket has a unique number. Each ticket sold for $1, and one ticket is drawn for a single prize of $75 million (and no other prizes). The expected value of a single ticket is A) $1. B) -$0.25. C) $75.

B) -$0.25. value x probability -1 x 1 = -1 75,000,000 x (1/100,000,000) = .75 -1 + .75 = .25

You are asked to create a 4-character password, and each character may be any of the 26 letters of the alphabet or the 10 numerals 0 through 9. How many different passwords are possible? A) 36 B) 36^4 C) 26^4 + 10^4

B) 36^4

Suppose that the probability of a hurricane striking Georgia in any single year is 1 in 10 and that this probability has been the same for the past 1000 years. Which of the following is implied by the law of large numbers? A) Georgia has been hit by exactly 100 hurricanes in the past 1000 years. B) Georgia has been hit by close to (but not necessarily exactly) 100 hurricanes in the past 1000 years. C) If no hurricanes have hit Georgia in the past 10 years, then the probability of a hurricane hitting next year is greater than 0.1.

B) Georgia has been hit by close to (but not necessarily exactly) 100 hurricanes in the past 1000 years. 1000 x (1/10) = 100 years

One person in a stadium filled with 100,000 people is chosen at random to win a free pair of airline tickets. What is the probability that it will not be you? A) 1 in 100,000 B) 0.99999 C) 0.99

B) 0.99999

An insurance company knows that the average cost to build a home in a new California subdivision is $100,000, and that in any particular year there is a 1 in 50 chance of a wildfire destroying all the homes in the subdivision. Based on these data and assuming the insurance company wants a positive expected value when it sells policies, what is the minimum that the company must charge for fire insurance policies in this subdivision? A) $50 per year B) $100,000 per year C) $2000 per year

C) $2000 per year 100,000 x(1/50) = $2000

The gambler's fallacy is the mistaken belief that a streak of bad luck makes a person "due" for a streak of good luck. Example (continued losses):Suppose you are playing the coin toss game, in which you win $1 for heads and lose $1 for tails. After 100 tosses you are $10 in the hole because you flip perhaps 45 heads and 55 tails. The empirical probability is 0.45 for heads.

So you keep playing the game. With 1000 tosses, you get 480 heads and 520 tails. Does the result agree with the law of large numbers? Have you gained back any of your losses? Explain The proportion of heads in your first 100 tosses was 45%. After 1000 tosses, the proportion of heads has increased to 480 out of 1000, or 48%. This agrees with the law of large numbers, because the proportion grew closer to 50%. However, after 1000 tosses, you've won $480 (for the 480 heads) while losing $520 (for the 520 tails), for a net loss of $40. In other words, your losses have actually increased from $10 to $40, despite the fact that the proportion of heads grew closer to 50%

How many seven-symbol license plates are possible if both numerals and upper-case letters can be used in any order?

With 10 numerals and 26 letters in the alphabet, there are 36 choices for each of the seven symbols. We are therefore selecting r = 7 symbols with n = 36 choices for each symbol, so the number of seven-symbol license plates is n^r = 36^7 = 78,364,164,096

Suppose you coach a team of four swimmers. How many different ways can you put together a four-person relay team? Note that this example involves selection from a single group in which each member of the group is selected exactly once. The order of arrangement matters, the team ABCD is different from the team DCBA.

You can choose any of the four swimmers for the first leg. Once you've chosen the first swimmer, you have three swimmers left to choose from for the second leg. Therefore, the number of possible choices for the first two legs combined is 4 x 3 =12 You then have two swimmers to choose from for the third leg, and only one choice for the last leg. The total number of possible arrangements for the relay is 4 x 3 x 2 x 1= 24

Based on an insurance company's past data, each year an average of 1/10 policyholders make a claim of $250, 1/150 policyholders make a claim of $5000, and 1/400 policyholders make a claim of $17,000. a. What would the insurance company have to charge per policyholder to break even? In other words, what would the premium have to be in order to make the expected value of the policy zero? b. If the company sells insurance policies for $300 annually, what is the expected profit or loss for the company per policyholder over the course of a year? c. If the company sold 325 policies, what would the overall expected profit or loss be?

a. (-250 x 1/10) = 25 (-5000 x 1/150) = 33.33 (-17,000 x 1/400) = 42.50 Add together = $101 to break even. b. $300 - $101 = $199 c. $325 x $199 = $64,729

Football teams have the option of trying to score either 1 or 2 extra points after a touchdown. They can get 1 point by kicking the ball through the goal posts or 2 points by running or passing the ball across the goal line. Suppose that in the NFL 1point kicks were successful 89% of the time, while 2point attempts were successful only 39% of the time. In either case, failure means zero points. a. Calculate the expected value of the 1point attempt: b. Calculate the expected value of the 2point attempt

a. (.89 x 1) + (.11 x 0) = .89 b. (.39 x 2 + (.61 x 0) = .78

You and some of your friends decide to visit the Eastern Idaho state fair. One of your friends comes to you and says that he wants you to play this new game he found. He says he just played it and won $10 after playing 3 times. Here are the game rules: You just pay $3 to start the game and spin a big spinner. Whichever number it lands on, you get that much money back. The spinner has four sections labeled 1, 2, 3 and 4 each is a different size. See the picture below of what the spinner looks like. You just finished taking FDMAT108 and remember learning about the expected value. You quickly calculate the expected value before you decide whether you will play or not. You find the probabilities of landing on each number and record them in a nice little table seen below. L(and on Number Probability 1 0.379 2 0.252 3 0.208 4 0.161 a. What would your expected earnings per game be? Round your answer to the nearest penny. b. Should you expect to get the "expected value" in the first game? c. What would your total expected earnings be if you played 100 times? Round your answer to the nearest penny.

a. .85 b. No C. .85 (-3 x 1) + (1 x .379) + (2 x .252) + (3 x .208) + (4 x .161) $.849 or $.85

Arrangements with Repetition

If we make r selections from a group of n choices, a total of n x n x n x ..... x n = n^r

Example: Suppose an automobile insurance company sells an insurance policy with an annual premium of $200. Based on data from past claims, the company has calculated the following probabilities: An average of 1 in 50 policyholders will file a claim of $2,000. An average of 1 in 20 policyholders will file a claim of $1,000. An average of 1 in 10 policyholders will file a claim of $500.Assuming that the policyholder could file any of the claims above, what is the expected value to the company for each policy sold?

Let the $200 premium be positive (income) with a probability of 1 since there will be no policy without receipt of the premium. The insurance claims will be negative (expenses). The expected value is 200 + (-2000) x 1/50 + (-1000) x 1/20 + (-500) x 1/20 = $60 This suggests that if the company sells many policies, then the return per policy, on average, is $60.)

A roulette wheel has 38 numbers: 18 black numbers, 18 red numbers, and the numbers 0 and 00 in green. Assume that all possible outcomes—the 38 numbers—have equal probability. a. What is the probability of getting a red number on any spin?

P(A) = 18/38 = .474 number of way red can occur divided by total number of outcomes.

A city has 12 candidates running for three leadership positions. The top vote-getter will become the mayor, the second vote-getter will become the deputy mayor, and the third vote-getter will become the treasurer. How many outcomes are possible for the three leadership positions?

The voters choose r = 3 leaders from a group of n = 12 candidates. The order matters because each position is different. 12P3 = 12!/(12 -3)! = 12!/9! = 12 x 11 x 10 = 1320

Suppose there are 25 students in your class. What is the probability that at least one person in the class has the same birthday as you?

Thi s is an at least once question. With 365 days a year, the probability that any particular student has your birthday is 1/365. Therefore, the probability that a particular student does not have your birthday is 364/365. P(at least one with your birthday) = 1 - [P(not your birthday)]^24 1-[364/365]^24 = .064 or about 6% or 1 in 6.

You coach a team of ten swimmers, from whom you must put together a four-person relay team. How many possibilities do you have in this case?

This time, you can choose one of ten swimmers for the first leg of the relay. Once you make the first choice, you have nine choices for the second leg, eight choices for the third leg, and seven choices for the fourth leg. 10 x 9 x 8 x 7 = 5040 Each of the 5040 relays represents a different permutation because each relay swimmer is selected from the same group of ten swimmers, no swimmer can swim more than once, and the order of the swimmers in the relay is important. However, in this case, the relays make use of only four of the total of ten swimmers. That is, the number of possible relays made from ten swimmers selected four at a time is 5040. 10 x 9 x 8 x 7 = 10 x 9 x 8 x 7 x 6!/ 6! = 10!/ (10 - 4)! = 5040 The number 10 in the numerator is the number of swimmers from which the coach may choose the four relay team, and, the number 10-4 = 6 in the denominator is the number of swimmers who do not swim in the relay.