Transcript 5.1

5.1: Randomness,
Probability and Simulation
Section 5.1
Randomness, Probability and Simulation
After this section, you should be able to…
 DESCRIBE the idea of probability
 DESCRIBE myths about randomness
 DESIGN and PERFORM simulations
The Idea of Probability
Chance behavior is unpredictable in the short run, but has a
regular and predictable pattern in the long run.
The probability of any
outcome of a chance
process is a number
between 0 (never
occurs) and 1 (always
occurs) that describes
the proportion of times
the outcome would
occur in a very long
series of repetitions.
The Law of Large Numbers
The law of large numbers says that if we observe more and
more repetitions of any chance process, the proportion of
times that a specific outcome occurs approaches a single
value.
Myths about Randomness
The myth of short-run regularity:
The idea of probability is that randomness is predictable in
the long run (1 million plus occurrences). Probability does
not allow us to make short-run predictions.
The myth of the “law of averages”:
Probability tells us random behavior evens out in the long
run. Future outcomes are not affected by past behavior.
Women have a 50% of having a boy with each pregnancy;
the gender of any previous children do not matter!
Performing a Simulation
The imitation of chance behavior, based on a model that accurately
reflects the situation, is called a simulation. Simulations are usually
done with a table of random digits, calculator random number
generator (RandInt) or computer software.
State: Identify the probability calculation at interest.
Plan: Describe how to use a chance device/tool to implement
one repetition of the process. Explain clearly how to identify
the outcomes of the chance process.
Do: Perform many (at least 20) repetitions of the simulation.
Conclude: Use the results of your simulation to answer the
question of interest, in context.
Performing a Simulation
For Example: What is the probability that a student earns an
80% on a true/false quiz written in Chinese? (Assume the
exam taker does not know any Chinese). Should the instructor
be concerned about cheating?
How can we simulate the probability of guessing 80% correct
on a True/False quiz?
Required Elements:
– State must include:
• Identify variable
• Statement of probability in symbols or words.
– Plan must include:
•
•
•
•
•
•
What tool?
What values are you assigning?
How many values are you picking each time?
How many times are you conducting the simulation?
What about repeat digits or ignored digits?
What are you recording?
Required Elements:
– Do must include:
• Simulation data, if number of trials is 20 or less
• Summary of data for larger trials
– Conclude must include:
• Statement of probability
• Answer to question
– Usually about being surprised/reasonable/expected, etc.
The Golden Ticket
At a local high school, 95 students have permission to park
on campus. Each month, the student council holds a “golden
ticket parking lottery” at a school assembly. The two lucky
winners are given reserved parking spots next to the school’s
main entrance. Last month, the winning tickets were drawn
by a student council member from the AP Statistics class.
When both golden tickets went to members of that same
class, some people thought the lottery had been rigged.
There are 28 students in the AP Statistics class, all of whom
are eligible to park on campus. Design and carry out a
simulation to decide whether it’s plausible that the lottery
was carried out fairly.
**See 5.1 WS
STATE:
• What is the probability that the lottery would
result in two winners from the AP Stats class?
• P (X=2), where x is the number of winners from
AP Stats
PLAN:
Using the table of random digits, we will randomly
assign each student a two digit number from 01 to 95.
We’ll label the students in the AP Statistics class from
01 to 28, and the remaining students from 29 to 95.
(Numbers from 96 to 00 will be skipped.) Starting at
the randomly selected row 139 and moving left to
right across the row, we’ll look at pairs of digits until
we come across two different values from 01 to 95.
These two values will represent the two students with
these labels will win the prime parking spaces. We will
record whether both winners are members of the AP
Statistics class (Yes or no). We will conduct the
simulation 18 times.
Required Elements:
– Plan must include:
• What tool?
– Table of Random of Digits, Calculator Random Number
Generator (RandInt), etc.
• What values are you assigning?
– 01 to 95
• How many values are you picking each time?
– 2 values
• How many times are you conducting the simulation?
– 18 times
• What about repeat digits or ignored digits?
– Ignore repeat digits within a single draw
• What are you recording?
– Yes for both AP Stats.
DO:
Students
Labels
AP Statistics Class
01-28
Other
29-95
Reading across row 139 in Table
D, look at pairs of digits until you
see two different labels from 0195. Record whether or not both
winners are members of the AP
Statistics Class.
Skip numbers from 96-00
55 | 58
89 | 94
04 | 70
70 | 84
10|98|43
56 | 35
69 | 34
48 | 39
45 | 17
X|X
X|X
✓|X
X|X
✓|Sk|X
X|X
X|X
X|X
X|✓
No
No
No
No
No
No
No
No
No
19 | 12
97|51|32
58 | 13
04 | 84
51 | 44
72 | 32
18 | 19
✓|✓
Sk|X|X
X|✓
✓|X
X|X
X|X
✓|✓
X|Sk|X
Sk|✓|✓
Yes
No
No
No
No
No
Yes
No
Yes
40|00|36 00|24|28
CONCLUDE:
Based on 18 repetitions of our simulation, both
winners came from the AP Statistics class 3
times, so the probability is estimated as
16.67%. Therefore is definitely possible for two
AP Stats students to be selected in a “fair”
drawing.
NASCAR
In an attempt to increase sales, a breakfast cereal company
decides to offer a NASCAR promotion. Each box of cereal will
contain a collectible card featuring one of these NASCAR
drivers: Jeff Gordon, Dale Earnhardt, Jr., Tony Stewart, Danica
Patrick, or Jimmie Johnson. The company says that each of the
5 cards is equally likely to appear in any box of cereal. A
NASCAR fan decides to keep buying boxes of the cereal until she
has all 5 drivers’ cards. She is surprised when it takes her 23
boxes to get the full set of cards. Should she be surprised?
Design and carry out a simulation to help answer this question.
STATE:
What is the probability of needing to buy 23 or
more cereal boxes to obtain one card from each
driver?
PLAN:
Using the calculator's random number generator
(RandInt) we are going to simulate 50 trials. We
will assign each driver a unique number 1 through
5. We will record how many trials it takes to get all
five values (drivers). We will record the total
number of digits required each time.
Driver
Label
Jeff Gordon
1
Dale Earnhardt, Jr.
2
Tony Stewart
3
Danica Patrick
4
Jimmie Johnson
5
DO:
Dotplot of 50 Trials
CONCLUDE:
We never had to buy more than 22 boxes to get
the full set of cards in 50 repetitions of our
simulation. Our estimate of the probability that it
takes 23 or more boxes to get a full set of driver is
roughly 0. Therefore, she should be surprised that
it took 23 cereal box purchases to find all 5 driver
cards.