Physics 353

Problems due Monday April 10.

Do problems 2.17, 2.19, 2.21 in Schroeder, Thermal Physics. 
These continue to explore the Einstein model of a solid.

Additional Problems:

Problem I is similar to Schroeder, problem 2.3. If you 
don't want to flip a coin 50,000 times, try a computer simulation! A 
Mathematica notebook is available on the course web site to help.

I.  Suppose that you flip 50 fair coins.
   1. How many possible outcomes (microstates) are there?
   2. How many ways are there of getting exactly 25 heads and 25 tails?
   3. What is the probability of getting exactly 25 heads and 25 tails?
   4. What is the probability of getting exactly 30 heads and 20 tails?
   5. What is the probability of getting exactly 10 heads and 40 tails?
   6. Plot a graph of the probability of getting n heads as a function of n.
   7. Try this once. How many heads did you get?
   8. Try this 1000 times. Plot the fraction of the time that you got n
      heads versus n and overlay your graph of the probability of getting 
      n heads as a function of n. 
   9. Suppose you thought that the coins were not fair, and wanted to 
      estimate the probability (p) of getting heads. How many times would 
      you have to toss the coins in order to be confident that your 
      estimate of p is correct? What do you mean by "confident"?

II. A year on Mars is 666 Mars days. Martians always celebrate their 
   birthdays, and are very excited if two people at the same gathering
   happen to have the same birthday (that is birth day, not necessarily
   birth day and birth year). In a random gathering of n Martians, what
   is the probability that two or more of them will have the same birthday? 
   Give a formula and make a graph of this probability versus n. 

Hint: it is a lot simpler to think about the probability that no two of the 
Martians have the same birthday. Then, consider the probability that no three
Martians have the same birthday, etc. If p is the probability that no Martians
have the same birthday, then (1-p) is the probability that two or more Martians
do have the same birthday.