Mathematical Modeling in Finance with Stochastic Processes

154 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES

(c) What is the approximate probability that the number of heads is

within 25 of the number of tails, that is, a difference of 5% or less

of the number of tosses in 500 tosses?

(d) Derive and then graph a simple power function that gives the

approximate probability that the number of heads is withinxof

the number of tails in 500 tosses, for 0≤x≤500.

2. (a) What is the probability that the number of heads is within 10
   of the number of tails, that is, a difference of 1% or less of the
   number of tosses in 1000 tosses?
   (b) What is the probability that the number of heads is within 10
   of the number of tails, that is, a difference of 0.5% or less of the
   number of tosses in 2000 tosses?
   (c) What is the probability that the number of heads is within 10
   of the number of tails, that is, a difference of 0.2% or less of the
   number of tosses in 5000 tosses?
   (d) Derive and then graph a simple power function that gives the
   approximate probability that the number of heads is withinxof
   the number of tails in 500 tosses, for 0≤x≤500.


3. Derive the rate, as a function ofn, that the probability of heads exceeds
   tails by a fixed valuesapproaches 1 asn→∞.


4. Derive the rate, as a function ofn, that the probability of heads exceeds
   tails by a fixed fractionapproaches 0 asn→∞.

Outside Readings and Links:

1. Virtual Laboratories in Probability and Statistics. Search the page for
   Random Walk Simulation and run the Last Visit to Zero experiment.


2. Virtual Laboratories in Probability and Statistics. Search the page for
   Ballot Experiment and run the Ballot Experiment.