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Probability Lab The Problem: Blue Mountain Property and Casualty Co. (BMP&C) sells, among other products, car insurance. Sue Mountie, a long standing customer, has had a claim in each of the last two years. Underwriting would like to know whether or not to cancel her insurance. An important piece of information: Underwriting doesn’t want to insure anyone who has a greater than 1 in 10 chance of having a claim in any given year. Underwriting has asked the actuarial department to determine if 2 consecutive years of claims over the life of the policy is consistent with their underwriting standards, assuming a 40 year life for the policy. In other words, what is the chance of having 2 or more consecutive years with significant claims over a 40 year period, if the probability of a significant claim in a given year is 1/10. What about 3 or more? We will solve this problem two ways: experimentally and analytically. Such experimental solutions are called “Monte Carlo” solutions because of the famous casino at Monte Carlo. The Monte Carlo Solution: We will use the “Random Number Generation” function in Excel to generate a random sequence of 40, 0’s and 1’s, for which the 0 is 9 times more likely than the 1. We interpret this sequence as representing the accident record of one individual over a 40 year period, where each 1 represents a year in which there was a claim. This sequence represents a “bad record” if it has 2 or more consecutive 1’s anywhere. We will generate 20 such sequences and count the number of “bad records” found, which will give an estimate of the probability someone having a bad record. This process may also be used to estimate the probability of 3 or more consecutive bad years. 1. Open an word document and put your name in the upper left. 2. Open an Excel spreadsheet 3. In Cell A1 type =randbetween(1,10) and fill down to A40- you will see a list of random numbers 4. Visually check to see if there are any consecutive 1’s- in the list? 5. To have excel check this we are going to employ the fact that the only way to get a sum of two is if each addend is 1. So in B2 type =A1+A2. Then drag this down to B40 6. Visually check are there any sums of 2 in your B column list? Do they correspond with consecutive 1s in the A columns? 7. Let’s ask excel to check for “2s” for us, in C2 type =IF(B2=2,1,0) this will return a 1 if the number is 2 and a zero if it is not 2. 8. In C42 type =IF(SUM(C2:C40)>=1,1,0), this will give us a 1 if there was one or more 2s in our list (or if the driver had two years with claims. 9. We have done this simulation just once, the power in simulation is doing it multiple times. 10. Highlight A1 to C42- drag it to column ET (this will give you 50 different simulations) 11. Now in A43 type =COUNT(B42:ET42) 12. Now in C43 type =(SUM(B42:BH42))/A43 This is the probability of having two consecutive claims. On your word document paste cells A1 to C43. Highlight cell C43 Answer the question: what is the probability of having two consecutive claims within 40 years in a complete sentence. Highlight your sentence. Repeat the process but now look for 3 consecutive claims. Do this on a new sheet. To make it 3 consecutive claims instead of 2, change B2 to be A1+A2+A3, and drag down. in C2 type =IF(B2=3,1,0). In your word document, Paste the first three columns from sheet 1 and highlight cell C43 again. Again, answer the question: what is the probability of having three consecutive claims within 40 years in a complete sentence. Highlight your sentence. Bonus: Can you solve these two problems analytically? How do they compare to your simulation?