Probability of coin and markov transition matrix

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SUMMARY

The discussion focuses on calculating probabilities related to flipping a fair coin and analyzing a Markov transition matrix. Specifically, it addresses the probability of obtaining a certain number of heads in 8992 flips, including conditions such as having more than half heads and less than a specific threshold. Additionally, it examines a Markov transition matrix defined by the parameters .5, .5, .75 - k, and .25 + k, where k is derived from dividing 8992 by 20000. The behavior of the system over multiple time steps is also analyzed, questioning whether the results will cycle, lead to extinction of a state, or converge to a limit value.

PREREQUISITES
  • Understanding of probability theory, particularly binomial distributions.
  • Familiarity with Markov chains and transition matrices.
  • Basic knowledge of limits and convergence in mathematical systems.
  • Ability to perform calculations involving large numbers and significant digits.
NEXT STEPS
  • Calculate the exact probability for the conditions outlined in the coin flip problem.
  • Explore the implications of the Markov transition matrix on long-term state behavior.
  • Investigate the concept of state extinction in Markov processes.
  • Learn about the application of large number approximations in probability theory.
USEFUL FOR

Mathematicians, statisticians, students studying probability and Markov processes, and anyone interested in advanced probability calculations and their applications.

hupdy
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1. Consider n flips of a fair coin. Calculate the probability:

a. n/2 < -Total number of heads

b. 5000 > total #heads

c. n/2 < total #heads < 5n/8

d. n < total #heads.

WHERE n = 8992

2. Consider the shopping problem
Markov transition matrix

.5 | .5
-----------------
.75 - k | .25 + k

where k = 8992 divided by 20000..

Start with initial v0 = (..5,.5) and describe the behavior of the
system for many time steps.

Does your result cycle, does one state become extinct, or does it
approach a limit value?Any help will be nice.
 
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Is this a homework question? If so, please post your attempts at a solution, before we can help you.
 
If you calculate the answer to #1a exactly, there are 2706 digits in the numerator and the same number in the denominator. :-p
 

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