Understanding
Randomness
Chapter XI
Rory Nimmons
Venkat Reddy
Main Concepts
Simulations are used for:
Can help to investigate a query for which there are many potential results,
whilst not intending to collect data, and the math is like super difficult to
do.
Main Concepts cont.
❏ Base simulation on random values that are to be
generated by a randomizing device or Internet
❏ Simulation model’s can provide useful insight on the
real world
Terms
Random: An event is random if we know what outcomes could happen,
but not which particular values will happen.
Random Numbers: Random numbers are hard to generate.
Nevertheless, several Internet sites offer an unlimited supply of equally
likely random values.
Terms
Simulation: A simulation models random events by using random
numbers to specify event outcomes with relative frequencies that
correspond to the true real-world relative frequencies we are trying to
model.
Simulation Component: The most basic result of a
component of a simulation is its outcome.
Terms
Outcome: An individual result of a component of a simulation is its
outcome
Trial: The sequence of several components representing events that we
are pretending will take place.
Response Variable: Values of the response variable record the
results of each trial with respect to what we were interested in.
Formula’s
For generating randomness❏ Internet Programs
❏ Calculator
❏ Textbook
Setting up a Simulation
Questions to ask yourself-
What is the component to be repeated?
How will you model the outcome?
How will you simulate the trial?
What is the response variable ?
How will you analyze the response variable?
Homework Question #5
Explain why each of the following simulations fails to model the real situation
properly.
A. Use a random integer from 0 through 9 to represent the number of heads
that appear when 9 coins are tossed
Answer: The outcomes are not equally likely; for example tossing 5 heads does
not have the same probability as tossing 0 or 9 heads, but the simulation
assumes they are likely equal
B. A basketball player takes a foul shot. Look at a random digit, using an odd
digit to represent a good shot and an even shot to represent a miss.
Answer: The even-odd assignment assumes that the players have an equally
likely chance of making or missing a shot, but in reality the likelihood of
making the shot depends on the player’s skill so
#5 continued
C. Use five random digits from 1 through 13 to represent the
denominators of the cards in a poker hand.
Answer: Suppose a hand had 4 aces. This might be
represented by 1,1,1,1, and any other number. The
likelihood for getting the first ace is different that for
getting for example the second or third, but with this
question the likelihood is the same.
Hw Q #15
Many states run lotteries to raise money. A Web site advertises that it know
“how to increase YOUR chances of Winning the Lottery.” They offer several
systems and criticize others as foolish. One system is called Lucky Numbers.
People who play the Lucky Numbers system just pick a “lucky” number to play,
but maybe some numbers are luckier than others. Let’s use a simulation to see
how well this system works. To make the situation manageable, simulate a
simple lottery in which a single digit from 0 to 9 is selected as the winning
number. Pick a single value to bet, such as 1, and keep playing it over and over.
You’ll want to run at least 100 trials. (If you can program the simulations on a
computer or programmable calculator, run several hundred. Or generalize the
questions to a lottery that chooses two-or three-digit numbers-for which you’ll
need thousands of trials.)
#15
A. What proportion of the time do you expect to
win?
Answer:You should win about 10% of the time
A. Would you expect better results if you picked
a “luckier” number, such as 77?
Answer: No, you should be able to win the same
rate with any number