Instructions: When you finish all problems, ask your TA to check your work.

Submit written responses.

Problem 1.

The following table represents different probabilities for some events. Events Prior Conditional Joint Posterior Probabilites  Probabilites Probabilities Probabilities
 

Problem 2 (The Monte Carlo Method). Essentially, the Monte Carlo method is a technique used to solve a numerical problem through the use of random sampling. It has applications  in physics, engineering, computer graphics, and statistics. Here, we will use the  Monte Carlo method to obtain an approximate solution to the following birthday problem:

What is the probability that (at least) 2  people in a group of 12 will have the same birthday?

(a) From page 106 of your textbook, what should the theoretical probability be   or this problem?

  (b) In cell A11, use the RANDBETWEEN function to randomly choose a number from the  set {1, 2, 3, . . . , 364,   65}, then using the fill handle, copy and paste this formula 11  times, from cell A12 to A22. This will be our first random sample   f size 12. Each  number in this sample will represent a birth date (excluding leap year birthdays),  and will simulate, say, a choosing a class of 12 students at random.

(c) Highlight the sample and using the fill handle, create 199 more of these random samples. Start from column B and end to column   GR. This will simulate, say, having 200 random classroom each filled with 12 students.

(d) Now, we need check if each of our samples have at least two identical numbers, i.e., if we have at least two people with the same birthday. Let us focus on our first sample in column A. Copy and paste the following formula into cell A25:IF(COUNTIF(A11:A22,A11)>1,1,0) This formula behaves as follows. The function COUNTIF will search for the value in cell A11 in the given range A11:A22. This count will always be at least 1 (why?). If result of COUNTIF is greater than 1 (indicating that there is a  matching valuesomewhere in the range), our formula will output a 1, otherwise, if there is no match, the output will be 0.

(e) Not only do we to check if any of our numbers in our sample match the first value, we need to check if we have matches with the other 11 numbers. So using the fill handle, copy and paste the same formula from cell A26 to A37. Be sure to use absolute referencing on the range if you decide to use the fill handle approach.

(f) Next, we need some indication that our sample actually has matching birthdays. In cell A37, write your own formula that indicates our sample has matching birthdays. (Hint: One way to do   this is to use the COUNTIF and SUM function together.) Apply step (e) and (f) to the other 199 samples by highlighting the  appropriate region and using the fill handle.

(g) Using the indicator functions constructed in the previous step, count the number of samples that have matching birthdays. Take this number and divide by 200, and we now have an approximation solution to the birthday problem above. Howclose is your  approximate answer to the theoretical answer? How might we obtain an even more accurate answer? Problem 3 (Simulating fair dice). In this section we will estimate the probability that the sum of two fair dice is seven.

1. What is the theoretical probability that the sum of two fair dice is seven?

2. Type =ROUNDDOWN(6*RND(),0)+1 in Cell E7, and fill in the range E7:F5006 with this formula. This formula will generate equally likely numbers from the set 1, 2, 3, 4, 5, 6.

We can interpret each random number as the outcome of tossing a die.

3. Type =E7+F7 in Cell G7. This number represents the result of summing the numbers on two fair dice. Fill in the range G7:G5006 with this formula.

4. Type =IF(G7=7,1,0) in Cell H7, and fill in the range H7:H5006 with this formula.

Here, a 1 means that the sum of the pair of dice is seven, and is 0 otherwise.

5. Type =average(H7:H7) in Cell I7, and fill in the range H7:H5006 with this formula.

6. Plot the values of column A (x-axis) against the values of column I (y-axis). Press the key F9 to recalculate the random numbers. What happens to your graph as the random numbers change? Is there anything that remains constant regardless of what random    numbers are chosen? Use the graph to estimate the probability that the sum of the numbers on two fair dice is seven.

7. How do your results compare to the theoretical probability?