Transcript A game of Chance

CHAPTER 6
PROBABILITY & SIMULATION
Section 6.1—SIMULATION
Section 6.1—Simulation
 Three methods of answering questions
about chance occurrences:
– Estimate based on actual observations of the
random phenomenon
– Develop a probability model and use it to
calculate theoretical answers.
– Start with a model that reflects the truth about
the random phenomenon and simulate many
repetitions to estimate the chances.
 What is simulation?
– It’s the imitation of chance behavior, based on a
model that accurately reflects the experiment
under consideration.
 Why is it good to use for finding the
likelihood of certain results?
– Once we have a good model or a trustworthy
way to simulate experiments, we can simulate
many repetitions quickly. Also, because if we
have more repetitions we’ll get a proportion of
repetitions that will be close to the true
population proportion. Simulation can give a
pretty good estimate of the probabilities.
How do we simulate?
 We can use random digits from a table,
graphing calculator, or computer
software to simulate many repetitions
quickly.
Simulation Steps
 Step 1: State the problem or describe the
experiment.
 Step 2: State the assumptions.
-There is no universal one, just basic
assumptions you need to make for a specific
scenario.
 Step 3: Assign digits to represent outcomes.
 Step 4: Simulate many repetitions.
-(Remember estimated proportions become
more accurate with more repetitions.)
 Step 5: Calculate relative frequencies and state
your conclusions.
A game of Chance
Find the Probability of winning the
game
The rules of the game
 A bag of marbles
contains 9 blue and 1
green marble.
 If you pull the green
marble from the bag
you win the game.
 You are given 2
chances to pick a
marble from the bag.
More Rules!
 If you pick the green marble on the first try
you win and stop playing the game.
 If you pick a blue marble, 4 additional blue
marbles are removed and you are given a
second chance.
 If you pick a green marble on this try you
win!
Determine the probability of winning
 Describe an appropriate random digit
assignment and perform 20 simulations.
 Begin on line 127 on the random digit table.
 Determine the experimental probability of
winning the game.
Step 1:
State the problem or describe the experiment.
There are 10 marbles: 9 blue, 1 green
UP TO two chances to win. Choosing green wins.
1st chance: 1 green in 10 total marbles—10% chance
2nd chance: 1 green in 5 total marbles—20% chance
What is the chance of winning this game?
Step 2:
State the assumptions.
We are assuming all of the marbles are the same
size and therefore have an equal chance of being
selected on any individual draw from the bag of
marbles.
Additionally, we are assuming that each selection
from the bag in a given round is independent from
another. Essentially, if I were to repeat the first
round over and over again, my first selection would
not impact my second selection.
Step 3:
Assign digits to represent outcomes.
We need two unique assignments: one for each round.
In each case, a single digit represents a pull of a
marble from the bag.
1st Attempt: 0green, win, game over
1-9blue, move on to second attempt
2nd Attempt: 0-1green, win, game over
2-9blue, lose, game over
Step 4:
Simulate many repetitions.
Looking at one or two digits at a time will simulate
one “play” of this game. If the first digit is a 0 then
the game is over and only one digit is needed. If the
first digit is not a 0 then a second digit is needed. So,
we will need at least 20 digits and at most 40 digits to
simulate playing the game 20 times.
4-3-L, 9-0-W, 9-9-L, 9-4-L, 7-7-L, 2-5-L, 3-3-L, 0-W,
6-4-L, 3-5-L, 9-4-L, 0-W, 0-W, 8-5-L, 1-6-L, 9-2-L,
5-8-L, 5-1-W, 1-7-L, 3-6-L
Step 5:
Calculate relative frequencies and state your
conclusions.
So, as we can see from our 20 simulations, we
would win 5 out of 20 times. We have an
experimental probability of 5/20 or 0.25. In other
words, based on our simulation, we would expect
someone to one this game roughly 25% of the time
they would play.
Do be more confident in our estimate, we would
want to simulate many more repetitions (known as
trials) which is easier to do with a computer. We
could establish an estimate closer to the theoretical
probability of winning which is 0.28.
Using the Calculator
 If you want to perform the same simulation
with the same digit assignments then type in
the following:
– randInt(0,9,100)L1:randInt(0,9,100)L2:((L1
=0) or (L2=0) or (L2=1))L3:sum(L3)/100
– When you do this, it is essentially repeating the
simulation 100 times. You can change the 100
in all three spots to change the number of
simulations. The number that is reported is the
proportion of times that the game is won.