Selected Answers to Problem Set 1

1. Many of you had the right idea for the calendar problem. Simply add up the days in each year between a set year and the year in question. Then add up the days in the months preceeding that month in question, then add up the days preceeding that day in question. Finding a specific date to start out with, and knowing January 1st for that year, makes this problem much easier. Many of you started with the current date; however, that day can and will change and it adds one more calculation--finding the number of days left in the year.

3. The smallest value of N so that the chance of rolling a pair of aces is greater than 50% is 25 rolls. Knowing that the chance of rolling a pair of aces is (1/36), the chance of not rolling a pair of aces is (35/36). Since this is probability with replacement, not rolling a pair of aces in N tries is equivalent to (35/36)^N. Therefore, the odds of rolling a pair of aces in N tries is 1 - (35/36)^N. We must find N when the odds are greater than 50%. When N is 24, then the odds are just under 50%. At N = 25, N is greater than 50%.

4c) Probability speaking, you are equally related to your aunt and to your brother's daughter. An easy way to look at this problem is to notice that you and your aunt share the same relationship as you and your brother's daughter. Genetically speaking, you share 50% of the genes of your mother/father, who in turn shares 50% of the genes with her/his sister. You also share 50% of the genes with your brother, who in turn shares 50% of his genes with his daughter. In both cases you are 25% similar to the persons in question.

4d) Probability speaking, you are more related to your uncle than to your first cousin's son. Similar to last problem, you share 25% of your genes with your uncle. As for your first cousin's son, your first cousin's father or mother is your uncle or aunt. Therefore, you share 25% with that specific aunt or uncle. That person's offspring will share 50% of that person, or 12.5%. That offspring's son, your first cousin's son, will share 50% once again, or 6.25%. Therefore, you share 6.25% of your genes with your first cousin's son which is much less than you share with your uncle.