Answer the following questions:

1. (Law of Large Numbers). Simulate the experiment: rolling of the two dice using XLSTAT or Excel to find the rolled sum.

Follow the instructions: a) Open a new excel document, b) Click on cell A1, then click on the function icon fx and select Math&Trig, then select RANDBETWEEN. c) In the dialog box, enter 1 for bottom and enter 6 for top. d) After getting the random number in the first cell, click and hold down the mouse button to drag the lower right corner of this first cell, and pull it down the column until 25 cells are highlighted. When you release the mouse button, all 25 random numbers should be present. e) Repeat the steps b) to d) for the second column. f) Put the rolled sum of two dice in the third column: Highlight the first two cells in the first row and click on AutoSum icon. Once you receive the sum of two values in the third cell, repeat the step d) to get rolled sum for all 25 values. g) Attach the screenshot of your Excel file to your assignment.

1-1. Find the (classical) probability that the rolled sum of two dice is equal to 9.

1-2. Based on the results of experiment (25 trials), estimate the probability (relative frequency approximation) that the rolled sum of two dice is equal to 9.

1-3. Repeat the simulation experiment for 50 and 100 trials, and estimate the probability (relative frequency approximation) that the rolled sum of two dice is equal to 9 for each experiment. Which probability has the closest value to the classical probability?

1-4. Briefly explain how these experiments demonstrate the Law of Large Numbers.

1-5. Construct the (classical) probability distribution of a rolled sum associated with the rolling of the two dice.

2. Men have XY or YX chromosomes and women have XX chromosomes. X- linked recessive genetic disease occurs when there is a defective X chromosome that occurs without a paired X chromosome that is good. A child with the xY or Yx (x represents a defective X chromosome) will have the disease and a child with XX, XY, YX, xX or Xx will not have the disease. Each parent contributes one of the two chromosomes to the child.

a.) If a father has XY chromosomes and the mother has Xx chromosomes, what is the probability that the son will inherit the disease?

b.) If a father has xY chromosomes and the mother has Xx chromosomes, what is the probability that the daughter will inherit the disease?

3. The probability that a visit to a primary care physician’s (PCP) office results in neither lab work nor referral to a specialist is 35%. Of those coming to a PCP’s office, 30% are referred to specialists (event A) and 40% require lab work (event B).

a.) Draw a Venn diagram and identify the following probabilities:

P(A), P(NOT A), P(B), P(NOT B), P(NOT A OR NOT B), P(A or B)

b.) Determine the probability that a visit to a PCP’s office results in both lab work and referral to a specialist.

c.) Determine the probability that a visit to a PCP’s office will result in lab work given that he/she has referred to a specialist.

4. A group of 22 students decided to play soccer, and they need to be divided into two teams of 11. Two people can be goalies, 4 people can play as forwards and the rest can play either defense or midfield.

a.) In how many ways can two forwards be chosen on each team if there are four forwards available?

b.) In how many ways can 22 students be divided into two equal groups if each team has one goalkeeper and two forwards?

5. There are 20 apples in a basket: 5 Macintosh and 10 Gala and 5 Spartan. You randomly select two apples.

a.) Develop a tree diagram of all possible outcomes of this experiment.

b.) Find the probability that the first apple is Gala and the second is Macintosh.

c.) Find the probability that at least one apple is Spartan.

6. (refer to the question 2) Suppose a father has YX chromosomes and the mother has Xx chromosomes.

*(Please note if you cannot answer part c.) of this question using xlstat then just answer it without using xlstat as you would manually)*

a.) Prove that the probability of a diseased child is 0.25.

b.) Construct a probability distribution of the numbers of diseased children if the family has three children.

c.) Find the mean and standard deviation number of diseased children manually and check the answer using XLSTAT.

d.) Is it a binomial distribution? Why or why not?

7. According to Statistics Canada (Share of Non-Alcoholic Beverage Market, 2008), 16% of Canadians consume coffee, 12% - tea, 72% - others. Twenty five people have registered for a workshop on potential health hazards associated with consumption of caffeine in food and dietary supplements.

*(Please note if you cannot answer part a.) and b.) of this question using xlstat then just answer it without using xlstat as you would manually)*

a.) Find the probability that exactly five people from the list of participants prefer drinking either tea or coffee. Calculate the probability manually and check your answer using XLSTAT.

b.) Find the probability that at least two people out of 25 prefer drinking coffee. Calculate your answer manually and check your answer using XLSTAT.

c.) What is the average number of people who prefer tea or coffee?

*Additional info for this question*

-Others 21%

-Carbonated Soft Drinks 16%

-Coffee 16%

-Milk 13%

-Tea 12%

-Bottled Water 11%

-Fruit Beverages 11%

**Subject Mathematics Advanced Statistics**