Professor Frank’s Math Blog: July 2020

Sunday, July 26, 2020

Probability Part 4 - Practice Problems and Answers (#18)

Problems:

In a lottery game, a player picks six numbers from 1 to 48. If 5 of the 6 numbers match those drawn, the player wins second prize. What is the probability of winning this prize?

Compute the probability that a 5-card poker hand is dealt to you that contains four aces?

A company estimates that 0.7% of their products will fail after the original warranty period but within two years of the purchase, with a replacement cost of $350. If they offer a 2 year extended warranty for $48, what is the company’s expected value of each warranty sold?

Answers:

In a lottery game, a player picks six numbers from 1 to 48. If 5 of the 6 numbers match those drawn, the player wins second prize. What is the probability of winning this prize? Answer: The number of possible outcomes is 48 choose 6 which is equal to 12, 271, 512. The number of ways to choose 5 out of the 6 winning numbers is 6 choose 5 which equals 6. The number of ways to choose 1 out of the 42 of the losing numbers is 42 choose 1 which is 42. By the basic counting rule, the number of favorable outcomes is: (6)(42) = 252. So, the probability of winning this prize is 252/12,271,512 = 21/1,022,626 which is approximately 0.0000205.

Compute the probability that a 5-card poker hand is dealt to you that contains four aces. Answer: The number of possible outcomes is 52 choose 5 which is equal to 2,598,960. Since there are four aces, there are 4 choose 4 equals 1 way to select four aces. Since there are 48 non-aces and we want one of them, there are 48 choose 1 = 48 ways to select one of the non-aces. By the Basic Counting Rule, there are (1)(48) = 48 ways to choose four aces and one non-ace. Thus, the probability of four aces is 48/2,598,960 = 1/54,145 which is approximately 0.0000185.

A company estimates that 0.7% of their products will fail after the original warranty period but within two years of the purchase, with a replacement cost of $350. If they offer a 2 year extended warranty for $48, what is the company’s expected value of each warranty sold? Answer: The outcome $48 - $350 = -$302 has a probability of 0.007. The outcome $48 has a probability of 0.993. The expected value is (-$302)(0.007) + ($48)(0.993) = $45.55.

Probability Part 4 - Notes

Here are the notes for this section. We will discuss probability using permutations and combinations, the birthday problem, and expected value. Let me know if you have any questions.

A 4 digit PIN number is selected. What is the probability that there are no repeated digits? Answer: Total possible pin numbers: (10)(10)(10)(10) = 10,000. No repeated digits means that all four digits have to be different and so there are 10 P 4 = 10!/(10 - 4)! choices. Thus, the probability is the # ways for no repeated digits / total # of PIN numbers = 10 P 4 / 10,000 = (10!/(10 - 4)!)/10,000 = 63/125.

Example: In a lottery, 48 balls numbered 1 through 48 are placed in a machine and 6 of them are drawn at random. If the six numbers drawn match the numbers that a player had chosen, the player wins $1,000,000. In this lottery, the order the numbers are drawn doesn't matter. Compute the probability that you win the million dollar prize if you purchase a single lottery ticket. Answer: # ways six numbers can be drawn is 48 choose 6 ways. There is only one winner so the probability is 1/(48 choose 6) = 1/12,271,512. Example: In the lottery from the previous example, if 5 of the 6 numbers drawn match the numbers that a player has chosen, the player wins a second prize of $1,000. Compute the probability that you win the second prize if you purchase a single lottery ticket. Answer: # possible outcomes = 48 choose 6, # ways to choose 5 of the 6 winning numbers is 6 choose 5, # ways to choose 1 out of the 42 losing numbers is 42 choose 1. Thus, the number of ways to win the second prize is given by the Basic Counting Rule: (6 choose 5) times (42 choose 1). So, the probability of winning second prize is (6 choose 5) times (42 choose 1) divided by 48 choose 6 which equals 21/1,022,626

Birthday Problem. Example: Suppose three people are in a room. What is the probability that there is at least one shared birthday among these three people? Answer: Probability if at least one birthday in common equals 1 - probability of no birthdays in common. We’ll find P(no shared birthday). Let the three people be denoted by X, Y, and Z. Assume that there are 365 days in a year. The probability that X and Y do not share a birthday is 364/365. The probability that X does not have the same birthday as Y or Z is 363/365. Thus, P(no shared birthday) = (364/365) times (363/365). Hence, P(shared birthday) = 1 - (364/365) which is approximately 0.0082.

Expected Value is the average gain or loss of an event if the procedure is repeated many times. We can compute the expected value by multiplying each outcome by the probability of that outcome, then adding up the products. Example: In a lottery, 48 balls numbered 1 through 48 are placed in a machine and 6 of them are drawn at random. If the six numbers drawn match the numbers that a player had chosen, the player wins $1,000,000. If they match 5 numbers, they will win $1,000. It costs $1 to buy a ticket. Find the expected value. Answer: P(matching all 6 numbers) = 1/12,271,512, P(matching all 5 numbers) = 252/12,271,512. The expected value is ($999,999)(1/12,271,512) + ($999)(252/12,271,512) + (-$1)(12,271,259/12,271,512) is approximately -$0.898. On average, one can expect to lose about 90 cents on a lottery ticket.

If the expected value of a game is 0, we call it a fair game. Example: A 40 yr. old man has a 0.242% risk of not surviving the next year. An insurance company charges $275 for a life insurance policy that pays a $100,000 death benefit. What is the expected value for the person buying the insurance? Answer: The expected value is ($99,275)(0.00242) + (-$275)(0.99758) which is approximately equal to -$33.

Probability Part 3 - Practice Problems and Answers (#17)

Problems:

At a restaurant you can choose from 3 appetizers, 8 entrees, and 2 desserts. How many different three course meals can you have?

A computer password must be eight characters long. How many passwords are possible if only the 26 letters of the alphabet are allowed.

In how many ways can first, second, and third prizes be awarded in a contest with 210 contestants?

Seven Olympic sprinters are eligible to compete in a 4 x 100 m relay race for the U.S.A Olympic team. How many four person relay teams can be selected from among the seven athletes?

Answers:

Probability Part 3 - Notes

Here are the notes for this section. We will learn about basic counting techniques, permutations, and combinations. Let me know if you have any questions.

Probability Part 2 - Practice Problems and Answers (#16)

Problems:

Compute the probability of drawing a King from a deck of cards and then drawing a Queen.

A jar contains 4 red marbles numbered 1 to 4 and 10 blue marbles numbered 1 to 10. A marble is drawn at random from the jar. Find the probability the marble is blue or even-numbered.

A jar contains 4 red marbles numbered 1 to 4 and 8 blue marbles numbered 1 to 8. A marble is drawn at random from the jar. Find the probability the marble is

a. Odd-numbered given that the marble is blue

b. Blue given that the marble is odd-numbered

A certain disease has an incidence rate of 0.3%. If the false negative rate is 6% and the false positive rate is 4%, compute the probability that a person who tests positive actually has the disease.

Answers:

Probability Part 2 - Notes

Here are the notes for this section. We will learn about conditional probability and Bayes’ Theorem. Let me know if you have any questions.

Probability Part 1 - Practice Problems and Answers (#15)

Problems:

A ball is drawn randomly from a jar that contains 6 red balls, 2 white balls, and 5 yellow balls. Find the probability of the given event.

Compute the probability of tossing a six-sided die (with sides numbered 1 through 6) and getting a 5.

Compute the probability of tossing a six-sided die and getting a 7.

A fair coin is flipped twice. What is the probability of showing heads on both flips?

Answers:

Probability Part 1 - Notes

Here are the notes for this section. We will study basic concepts and working with events in probability. Let me know if you have any questions.

Statistics Part 3 & 4 - Practice Problems and Answers (#14)

Problems:

The table below shows scores on a Math Test:

80 50 50 90 70 70 100 60 70 80 70 50
90 100 80 70 30 80 80 70 100 60 60 50

a. Complete the frequency table for the Math Test scores

b. Construct a histogram of the data

c. Construct a pie chart of the data

A group of adults were asked how many children they have in their families. The bar graph below shows the number of adults who indicated each number of children.

a. How many adults were questioned?

b. What percentage of the adults questioned had zero children?

Refer back to the histogram from the previous question.

a. Compute the mean number of children for the group surveyed

b. Compute the median number of children for the group surveyed

c. Write the 5-number summary for this data.

d. Create a box plot

Answers:

Statistics Part 4 - Notes

Here are the notes for this section. We will learn about measures of central tendency in this section. Let me know if you have any questions.

Statistics Part 3 - Notes

Here are the notes for this section. We will learn about presenting categorical data graphically in this section. Let me know if you have any questions.

Statistics Part 2 - Practice Problem and Answer (#13)

Problem: For the clinical trials of a weight loss drug containing Garcinia cambogia the subjects were randomly divided into two groups. The first received an inert pill along with an exercise and diet plan, while the second received the test medicine along with the same exercise and diet plan. The patients do not know which group they are in, nor do the fitness and nutrition advisors.

a. Which is the treatment group?

b. Which is the control group?

c. Is this study blind, double-blind, or neither?

d. Is this best described as an experiment, a controlled experiment, or a placebo controlled experiment?

Answer:

Saturday, July 25, 2020

Statistics Part 2 - Notes

Here are the notes for this section. We will learn about how to mess things up before we start in statistics as well as experiments. Let me know if you have any questions.

Statistics Part 1 - Practice Problems and Answers (#12)

Problems:

The city of Raleigh has 9500 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 350 randomly selected registered voters was conducted. One hundred and twelve said they’d vote for Brown, 238 said they’d vote for Feliz, and 31 were undecided.

a. What is the population of this survey?

b. What is the size of the population?

c. What is the size of the sample?

d. Give the sample statistic for the proportion of voters surveyed who said they’d vote for Brown.

e. Based on this sample, we might expect how many of the 9500 voters to vote for Brown?

Identify the most relevant source of bias in this situation: A survey asks people to report their actual income and the income they reported on their IRS tax form.

In a study, you ask the subjects their age in years. Is this data qualitative or quantitative?

Does this describe an observational study or an experiment: The temperature on randomly selected days throughout the year was measured.

In a study, the sample is chosen by separating all cars by size, and selecting 10 of each size grouping. What is the sampling method?

Answers:

Statistics Part 1 - Notes

Here are the notes for this section. We will discuss: populations and samples, categorizing data, and sampling methods. Let me know if you have any questions.

Finance Part 4 - Practice Problems and Answers (#11)

Problems:

Pat deposits $6,000 into an account earning 4% compounded monthly. How long will it take the account to grow to $10,000?

Chris has saved $200,000 for retirement and it is in an account earning 6% interest. If she withdraws $3,000 a mont, how long will the money last?

Answers: