Math 210 Spring 2019  

Problem Set 1, Due Tues. Jan. 29 in Class
(late papers OK until 1:00 Wednesday)

Please put your papers in the mailbox (DRL 4W3) of our TA Yansong Gao

1. Calendar Problems
Given a date, such as March 17, 2011, determine the day of the week (Monday, Tuesday, etc.). [Hint: If today is a Saturday, a day 700 days in the future is also a Saturday, while a day 702 days in the future is "clearly" a Monday (why?)].
Note: A year is a leap year if the year is divisible by 4 and it is not divisible by 100 -- unless it is also divisible by 400. Thus the year 1900 was not a leap year but 2000 was.
To check your procedure, the Unix command cal   2176 gives a calendar for the year 2176.

Probability Problems: The following use only intuitive concepts from probability. We'll discuss them a bit in class on Tuesday. Speak with your friends.
As a reference see, for instance, Grinstead and Snell, Introduction to Probability, Chapter 1 and Chapter 2 .

2. Genetically, to whom are you more closely related:
a). Your brother or your mother?
b). Your granddaughter or your sister's son?
c). Your aunt or your brother's daughter?
d). Your uncle or your first cousin's son?
e). In a court case, say there the deceased has no surviving children, grand children, ... (direct lineal descendants) and no will. But there are aunts, uncles, and cousins. The estate has lots of money. What algorithm might the Court use to distribute the legacy?

3. Say you toss a pair of dice N times. You win if a pair of 1's appear on the same toss. What is the smallest value of N so your likelihood of winning is more than 50%?

4. In an office, say your personal code to use the photo copy machine is the last four digits of your social security number.
a). If 100 people share that copier, what is the probability that at least two people have the same code?
b). Same question with 150 people.

5. A well-known folk legend. On the night before the big chemistry exam, two students were partying all night -- and over slept the exam. Their excuse to the professor was that they were driving from some friends who lived far away and had a flat tire; could they please take a make-up exam. The professor agreed. She wrote an exam and put them in different rooms to take it. The first question, worth 5 points, was easy. The second question, on the other side of the page was worth 95 points. It simply asked, "Which tire was it?".
What is the probability that both students would give the same answer?

6. Half the females and one-third of the males in a class are smokers. Also, two-thirds of the students are male. What fraction of the smokers are female?