Probability of coin and markov transition matrix

In summary, the conversation discusses probability calculations for flipping a fair coin and a shopping problem involving Markov transition matrices. The first part involves calculating the probability of certain outcomes based on the number of coin flips and the second part involves analyzing the behavior of the system over time.
  • #1
hupdy
8
0
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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  • #2
Is this a homework question? If so, please post your attempts at a solution, before we can help you.
 
  • #3
If you calculate the answer to #1a exactly, there are 2706 digits in the numerator and the same number in the denominator. :tongue2:
 

1. What is the probability of getting heads or tails when flipping a coin?

The probability of getting heads or tails when flipping a coin is equal, assuming the coin is fair. This means that the probability of getting heads is 0.5, or 50%, and the probability of getting tails is also 0.5.

2. How do you calculate the probability of a specific outcome with a coin toss?

To calculate the probability of a specific outcome with a coin toss, you would divide the number of desired outcomes by the total number of possible outcomes. For example, if you want to know the probability of getting heads, which is one possible outcome out of two total outcomes, the probability would be 1/2 or 0.5.

3. What is a Markov transition matrix?

A Markov transition matrix is a mathematical tool used to represent the probabilities of transitioning from one state to another in a system. It is commonly used in analyzing and predicting the behavior of complex systems, such as economics, weather patterns, and biological processes.

4. How do you use a Markov transition matrix to calculate the probability of future events?

To calculate the probability of future events using a Markov transition matrix, you would first determine the current state of the system. Then, you would use the transition probabilities in the matrix to calculate the probability of transitioning to each possible future state. The sum of these probabilities would give you the overall probability of each future event.

5. Can the probability of a future event be accurately predicted using a Markov transition matrix?

While a Markov transition matrix can provide valuable insights into the probabilities of future events, it is not a perfect predictor. The accuracy of the predictions depends on the accuracy of the transition probabilities and the stability of the system. Additionally, external factors and unforeseen events can also impact the accuracy of the predictions.

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