### Experimental Probability and Simulation ppt

``` A simulation
imitates a real situation
 Is supposed to give similar results
 And so acts as a predictor of what should
actually happen
 It is a model in which repeated
experiments are carried out for the
purpose of estimating in real life
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
Used to solve problems using experiments when it
is difficult to calculate theoretically
Often involves either the calculation of:
◦ The long-run relative frequency of an event happening
◦ The average number of ‘visits’ taken to a ‘full-set’

Often have to make assumptions about situations
being simulated. E.g. there is an equal chance of
producing a boy or a girl
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Maths online
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AC/on
RUN <Exe>
OPTN
F6
PROB
Ran#
To Simulate tossing of a coin
1.
◦
Ran#


Tails: 0.500 000 – 0.999 999
To simulate LOTTO balls
2.
◦
◦
1+40Ran#, truncate the result to 0 d.p., or
0.5+40Ran#, truncate the result to 0 d.p.
3. To simulate an event which has 14% chance of
success
◦
100Ran#, truncate the result to 0 d.p.

◦
0 – 13 for success, 14-99 for failure, or
1+100Ran#, truncate the result to 0 d.p.

1-14 for success, 15-100 for failure
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Assume each day has equal
probability (1/7)
RANDBETWEEN(1,7)
Generate 4 random numbers
to simulate one family
Repeat large number of times
Day of
the week
Random
Number
Sunday
1
Monday
2
Tuesday
3
Wednesda 4
y
Thursday
5
Friday
6
Saturday
7
The description of a simulation should contain at least the
following four aspects:
Tools


Definition of the probability tool, eg. Ran#, Coin, deck of
cards, spinner
Statement of how the tool models the situation
Trials


Definition of a trial
Definition of a successful outcome of the trial
Results


Statement of how the results will be tabulated giving an
example of a successful outcome and an unsuccessful
outcome
Statements of how many trials should be carried out
Calculations
Statement of how the calculation needed for the conclusion
will be done
Number of ‘ successful ’ results
Long-run relative frequency =
Number of trials
Mean =
Number of ‘ successful
Number of trials
’ results
Tool: First digit using calculator 1+10Ran#
Odd Numbers stands for ‘Boy’ and
Even Number stands for ‘Girl’
Trial: One trial will consist of generating 4 random numbers to
simulate one family.
A Successful trial will have 2 odd and 2 even numbers.
Results:
Trial
Outcome of
Result of trial
trial
1
2357
Unsuccessful
2
4635
Successful
Number of Trials needed: 30 would be sufficient
Calculation:
Number of ‘ successful ’ results
Probability of 2 boys & 2 girls =
Number of trials
Tool: Generate random numbers between 1 & 6 (inclusive), each number
stands for each toy.
Trial: One trial will consist of generating random numbers till all
numbers from 1 to 6 have been generated.
Count the number of random numbers need to get one full set
Results:
Trial Toy
1
Toy
2
Toy
3
Toy
4
Toy5
Toy6
1
Y
Y
Y
Y
Y
Y
10
2
Y
Y
Y
Y
Y
Y
19
Number of Trials needed: 30 would be sufficient
Calculation:
Average number of visits =
Total visits
Number of trials
Tally
Total
Visits
Tool: The probability that Mary guesses a question true is one
half.
First digit using calculator 1 + 10Ran#
1to 5 stands for ‘correct answer’
6 to 10 stands for ‘incorrect answer’
Trial: One trial will consist of generating 3 random numbers to
simulate Mary answering one complete test.
A successful outcome will be getting atleast 2 of the 3
random numbers between 1 and 5.
Results: Trial
Outcome of Trial
Result of Trial
1
122
Successful trial
2
167
Unsuccessful trial
Number of Trials needed: 30 would be sufficient
Calculation: Estimate of probability of ‘passing’ the exam =
Number of ‘ successful
Number of trials
’ results
Tool:
The probability that Mary guesses a question true is one half.
First digit using calculator 1 + 10Ran#
1to 5
6 to 10 stands for ‘incorrect answer’
Trial: One trial will consist of generating 8 random numbers to simulate
A successful outcome will be getting atleast 4 of the 8 random numbers
between 1 and 5.
Results: Trial
Outcome of Trial Result of Trial
1
12236754
Successful trial
2
13672987
Unsuccessful trial
Number of Trials needed: 30 would be sufficient
Calculation:
Number
Estimate of probability of ‘passing’ the exam =
of ‘ successful
Number of trials
’ results
Problem:
Lotto 40 balls and to win you must select 6 in any order.
In this mini Lotto, there are only 6 balls and you win
when you select 2 numbers out of the 6.
Design and run your own simulation to estimate the
probability of winning (i.e. selecting 2 numbers out of
the 6)
Calculate the theoretical probability of winning.
Tool:
4)
Trial:
Results:
Two numbers (between 1 and 6) will need to be selected first (say 2 &
First digit using calculator 1 + 6Ran#, ignore the decimals.
One trial will consist of generating 2 random numbers
A successful outcome will be getting 2 of the 6 random numbers
generated
Trial
Outcome of Trial
Result of Trial
1
24
Successful trial
2
13
Unsuccessful trial
Number of Trials needed: 50 would be sufficient
Calculation:
Estimate of probability of ‘winning’ = Number of ‘successful’ outcome
Number of trials
Theoretical probability in this case is 1/15
```