Statistical simulation technique that provides approximate solution to problems expressed mathematically. It utilize the sequence of random number to perform the simulation.

1. MONTE
CARLO
SIMULATION

2. Submitted By :
 Ayush Singh (14/ics/016)
 Deepak Singh (14/ics/017)
 Deepti Singh (14/ics/018)
 Dusyant Pratap Singh (14/ics/020)
 Jaya Koli (14/ics/022)

3. MONTE CARLO SIMULATION:
 Statistical simulation technique that provides
approximate solution to problems expressed
mathematically.
 It utilize the sequence of random number to perform
the simulation.

4.  The technique is used by professionals in widely
disparate fields such as
1. Finance
2. Project Management
3. Energy, Manufacturing
4. Engineering
5. Research and development
6. Insurance
7. Oil and gas and the environment.

5. WHY MONTE CARLO SIMULATION?
 Easy and Efficient
 Simple implementation on computer.
 Furnishes the decision-maker with a range of
possible outcomes and the probabilities that will
occur for any choice of action.

6. HISTORY:
 Monte carlo simulation was named after the city in
Monaco. The name is a reference to a famous casino
in Monaco where Ulam’ uncle would borrow money
to gamble.
 The technique was first used by the scientists
working on the atom bomb during second world war.
 1930’s: Enrico Femi uses Monte Carlo in the
calculation of neutron diffusion.

7.  1940’s: Stan Ulam while playing solitaire tries to
calculate the likelihood of winning based on the
initial layout of cards.
 1950’s: Many papers on monte carlo simulation
appeared in physics literature. The first major
MCMC(Markov chain Monte Carlo) paper was
published by metropolis et al in 1953.
 1951:Ulman is primarily known for designing the
hydrogen bomb with Edward Teller.

8. EXAMPLE: Estimating the value of Pi

10. MONTE CARLO SIMULATION TECHNIQUE
 It is an experiment on choice.
 Uses random number and require decision making
under uncertainties.

11. Understanding of this technique
 To understand this technique this is break down in 5
steps.
1. Establishing probability Distribution
2. Cumulative probability Distribution
3. Setting random number Intervals
4. Generating Random number
5. To find the answer of question asked using the
above four step.

12. Question
 Ram is a dentist who schedule all his patients for 30
minutes appointment. Some of the patient take more
or less then 30 minutes depending on the type of
dental work to be done. The following summary
show the various categories of work, their
probabilities and the time actually needed to
complete the work :-

13. Category Time Require No of Patients Probability
Filling 45 Min 40
Crown 60 Min 15
Cleaning 15 Min 15
Extracting 45 Min 10
Check Up 15 Min 20

14.  Probability =
Favourable Outcome
Total Outcomes

15. Category Time Require No of Patients Probability
Filling 45 Min 40 0.40
Crown 60 Min 15 0.15
Cleaning 15 Min 15 0.15
Extracting 45 Min 10 0.10
Check Up 15 Min 20 0.20

16.  Simulate the Dentist’s Clinic for four Hour and find
out the average waiting time for the patients as well
as the idleness of the doctor. Assume that all the
patients show up at the clinic at exactly their
scheduled, arrival time starting at 8:00 am .
 Use the following random numbers for handling the
above problem.
 40, 82, 11, 34, 25, 66, 17, 79.

17. Category Probability Cumulative
Probability
Random
Number
Filling 0.40
Crown 0.15
Cleaning 0.15
Extracting 0.10
Check Up 0.20

18. Category Probability Cumulative
Probability
Random
Number
Filling 0.40 0.40 00-39
Crown 0.15 0.55 40-54
Cleaning 0.15 0.70 50-69
Extracting 0.10 0.80 70-79
Check Up 0.20 1.00 80-99

19. Patient Schedule
Arrival
Random
Number
Category Service
Time needed
1. 8:00
2. 8:30
3. 9:00
4. 9:30
5. 10:00
6. 10:30
7. 11:00
8. 11:30

20. Patient Schedule
Arrival
Random
Number
Category Service
Time needed
1. 8:00 40 Crown 60 Min
2. 8:30 82 Check Up 15 min
3. 9:00 11 Filling 45 min
4. 9:30 34 Filling 45 min
5. 10:00 25 Filling 45 min
6. 10:30 66 Cleaning 15 min
7. 11:00 17 Filling 45 min
8. 11:30 79 Extracting 45 min

21. Final Table for answer
Patient Schedul
e
Arrival
Service
Start
Service
Duratio
n
Service
End
Waiting
(in
minutes
)
Idle
Time
1. 8:00
2. 8:30
3. 9:00
4. 9:30
5. 10:00
6. 10:30
7. 11:00
8. 11:30

22. Patient Schedul
e
Arrival
Service
Start
Service
Duratio
n
Service
End
Waiting
(in
minutes
)
Idle
Time
1. 8:00 8:00 60 min 9:00 0 0
2. 8:30 9:00 15 min 9:15 30 0
3. 9:00 9:15 45 min 10:00 15 0
4. 9:30 10:00 45 min 10:45 30 0
5. 10:00 10:45 45 min 11:30 45 0
6. 10:30 11:30 15 min 11:45 60 0
7. 11:00 11:45 45 min 12:30 45 0
8. 11:30 12:30 45 min 1:15 60 0
Total 285

23.  Average Waiting Time =
285
8
= 35.625 min

24.  Monte Carlo simulation is a technique used to study how
a model responds to randomly generated inputs. It
typically involves a three-step process:
1.)Randomly generate “N” inputs (sometimes called
scenarios).
2.)Run a simulation for each of the “N” inputs.
Simulations are run on a computerized model of the
system being analysed.
3.)Aggregate and assess the outputs from the
simulations. Common measures include the mean value
of an output, the distribution of output values, and the
minimum or maximum output value.

25.  The MATLAB® language provides a variety of high-
level mathematical functions you can use to build a
model for Monte Carlo simulation and to run those
simulations. MATLAB is used for financial
modelling, weather forecasting, operations analysis,
and many other applications.

26.  Problem-To estimate the probability that 5 cards drawn of a
deck of 52 cards are hearts.
Sol:-(code)
%therotical p=0.00049116
number_of_runs=100000;
allHeart = zeros(1,number_of_runs);
for n=1:number_of_runs
deck=randperm(52);
number_of_cards_drawn=5;
draw = deck(1:number_of_cards_drawn);
count=0;

27.  for i=1:number_of_cards_drawn
if draw(i)<=13;
count = count+1;
end
end
if count == number_of_cards_drawn
allHeart(n)=1;
else
allHeart(n)=0;
end
end
S=sum(allHeart);
p=S/number_of_runs;

28. Applications
 Use in Mathematics
 The Monte Carlo methods are used in mathematics
to solve various problems by generating suitable
random numbers and observing that fraction of the
numbers that obeys some property or properties.
 The method is useful for obtaining numerical
solutions to problems too complicated to solve
analytically. The most common application of the
Monte Carlo method is Monte Carlo integration.

29. Integration
 Deterministic numerical integration algorithms
work well in a small number of dimensions, but
encounter two problems when the functions have
many variables.
1. The number of function evaluations needed
increases rapidly with the number of dimensions.
For example, if 10 evaluations provide adequate
accuracy in one dimension, then 10^100 points are
needed for 100 dimensions—far too many to be
computed. This is called the curse of dimensionality

30. 2. The boundary of a multidimensional region may be
very complicated, so it may not be feasible to reduce
the problem to an iterated integral. 100 dimensions is
by no means unusual, since in many physical
problems, a "dimension" is equivalent to a degree of
freedom.

31.  Monte Carlo methods provide a way out of this
exponential increase in computation time. As long as
the function in question is reasonably well-behaved,
it can be estimated by randomly selecting points in
100-dimensional space, and taking some kind of
average of the function values at these points. By the
central limit theorem,
 this method displays 1/ sqrt(N) convergence—i.e.,
quadrupling the number of sampled points halves
the error, regardless of the number of dimensions.

32.  A refinement of this method, known as importance
sampling in statistics, involves sampling the points
randomly, but more frequently where the integrand
is large. To do this precisely one would have to
already know the integral, but one can approximate
the integral by an integral of a similar function or use
adaptive routines such as stratified sampling,
recursive stratified sampling, adaptive umbrella
sampling or the VEGAS algorithm.

33.  A similar approach, the quasi-Monte Carlo method,
uses low-discrepancy sequences.
 These sequences "fill" the area better and sample
the most important points more frequently, so
quasi-Monte Carlo methods can often converge on
the integral more quickly.

34.  Another class of methods for sampling points in a
volume is to simulate random walks over it (Markov
chain Monte Carlo). Such methods include the
Metropolis-Hastings algorithm, Gibbs sampling,
Wang and Landau algorithm, and interacting type
MCMC methodologies such as the sequential Monte
Carlo samplers.

35.  Monte-Carlo integration works by comparing
random points with the value of the function

36.  Errors reduce by a factor 1/sqrt(N).

37. Simulation and optimization
 The problem is to minimize (or maximize) functions
of some vector that often has a large number of
dimensions.
 Many problems can be phrased in this way: for
example, a computer chess program could be seen
as trying to find the set of, say, 10 moves that
produces the best evaluation function at the end.
 In the traveling salesman problem the goal is to
minimize distance traveled

38.  The traveling salesman problem is what is called a
conventional optimization problem.
 That is, all the facts (distances between each
destination point) needed to determine the optimal
path to follow are known with certainty and the goal
is to run through the possible travel choices to come
up with the one with the lowest total distance.

39. Inverse problems
 Probabilistic formulation of inverse problems leads
to the definition of a probability distribution in the
model space.
 This probability distribution combines prior
information with new information obtained by
measuring some observable parameters (data).
 As, in the general case, the theory linking data with
model parameters is nonlinear, the posterior
probability in the model space may not be easy to
describe.

40.  When analyzing an inverse problem, obtaining a
maximum likelihood model is usually not sufficient,
as we normally also wish to have information on the
resolution power of the data.
 In the general case we may have a large number of
model parameters, and an inspection of the
marginal probability densities of interest may be
impractical, or even useless.

41.  But it is possible to pseudo randomly generate a
large collection of models according to the posterior
probability distribution and to analyze and display
the models in such a way that information on the
relative likelihoods of model properties is conveyed
to the spectator.
 This can be accomplished by means of an efficient
Monte Carlo method, even in cases where no explicit
formula for the a priori distribution is available.

42.  The best-known importance sampling method, the
Metropolis algorithm, can be generalized, and this
gives a method that allows analysis of (possibly
highly nonlinear) inverse problems with complex a
priori information and data with an arbitrary noise
distribution.

43. Advantages of Monte Carlo simulation
 Probabilistic Results. Results show not only what
could happen, but how likely each outcome is.
 Graphical Results. Because of the data a Monte Carlo
simulation generates, it’s easy to create graphs of
different outcomes and their chances of occurrence.
This is important for communicating findings to
other stakeholders.

44. Advantages of Monte Carlo simulation
 Sensitivity Analysis. With just a few cases,
deterministic analysis makes it difficult to see which
variables impact the outcome the most.
In Monte Carlo simulation, it’s easy to see which
inputs had the biggest effect on bottom-line results

45. Advantages of Monte Carlo simulation
 Scenario Analysis. In deterministic models, it’s very
difficult to model different combinations of values
for different inputs to see the effects of truly different
scenarios.
Using Monte Carlo simulation, analysts can see
exactly which inputs had which values together when
certain outcomes occurred.
This is invaluable for pursuing further analysis.

46. Advantages of Monte Carlo simulation
 Correlation of Inputs. In Monte Carlo simulation, it’s
possible to model interdependent relationships
between input variables.
It’s important for accuracy to represent how, in
reality, when some factors goes up, others go up or
down accordingly.

47. Disadvantages of Monte Carlo simulation
 Like most methods based on probability theory,
Monte Carlo methods are data‐intensive.
 Consequently, they usually cannot produce results
unless a considerable body of empirical information
has been collected, or unless the analyst is willing to
make several assumptions in the place of such
empirical information.

48. Disadvantages of Monte Carlo simulation
 Although appropriate for handling variability and
stochasticity, Monte Carlo methods cannot be used
to propagate partial ignorance under any frequentist
interpretation of probability.
 Monte Carlo methods cannot be used to conclude
that exceedance risks are no larger than a particular
level.
 Finally, Monte Carlo methods cannot be used to
effect deconvolutions to solve backcalculation
problems such as often arise in remediation
planning.

49. Conclusion
 We conclude that :
 Montecarlo simiulation has an Easy,Efficient and
Simple implementation on computer.
 Furnishes the decision-maker with a range of
possible outcomes and the probabilities that will
occur for any choice of action.
 It is an experiment on choice.
 Uses random number and require decision making
under uncertainties.

50. Conclusion
 Using Monte Carlo simulation, analysts can see
exactly which inputs had which values together when
certain outcomes occurred.
 Monte Carlo methods are data‐intensive.
 Monte Carlo methods cannot be used to effect
deconvolutions to solve backcalculation due to
remediation planning .

51. Future Scope
 In such models, uncertainty and incomplete knowledge is
dealt with by sampling probabilistically over the space of
possible trajectories ("Monte Carlo simulation").
 . We argue that such simulation-based models are too
weak, in that there are many important aspects of human
physical reasoning that cannot be carried out this way, or
can only be carried out very inefficiently; and too strong,
in that humans make large systematic errors that the
models cannot account for.
 We conclude that simulation-based reasoning makes up
at most a small part of a larger system that encompasses
a wide range of additional cognitive processes.