Studied by 0 people
Looks like no one added any tags here yet for you.
Purpose of a simulation (5.1)
It is a powerful method for modeling chance behavior
what is unpredictable in the short term but becomes regular and predictable in the long term? (5.1)
chance behavior; probability
What does probability explain: (5.1)
it describes what happens in very many trials
Law of large numbers (5.1)
If we observe many repetitions of any chance process, the proportion of times that a specific outcome approaches a single value. (that single value is probability)
Definition of probability: (5.1)
The probability of any outcome of a chance process is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions.
- gives us a language to describe the long-term regularity of random behavior.
Myth of the future outcomes (5.1)
The myth is that it must make up for an imbalance of the value that has been like six or (any number) straight of an outcome
- that's is not true b/c they have an equal chance regardless. It's just an illusion.
Four step process for performing a simulation (5.1)
State: ask a question of interest about some chance process
Plan: Explain how to use a chance device to imitate one repetition if the process. Tell what you will record at the end of each repetition
Do: Perform many repetitions of the simulation
Conclude: Use the results of your simulation to answer the question of interest.
Using the four step framework for the "1 in 6 wins" game (5.1)
State: What's the probability that 3 or more of 7 people who buy a 20-ounce bottle of soda win a prize if each bottle has a 1/6 chance of saying, "You're a winner!"?
Plan: Use a six-sided die to determine the outcome for each person's bottle of soda
- roll the die seven times, once for each person
- Record the number of people who win a prize
Do: have each student perform several repetitions
Conclude: Out of 125 total repetitions of the simulation, there were 15 times where three or more of the seven people won a prize. So our estimate of the probability is 15/125, or about 12%. It seems plausible that the company is telling the truth.
What type of integers are we looking for when using either a RDT or randInt (5.1)
First unique x numbers that fit the condition and stay within the interval.
NASCAR CARDS AND CEREAL BOXES EXAMPLE (5.1)
Problem: Each box of cereal will contain a collectible card featuring one of these NASCAR drivers: Jeff Gordon, Dale Earnhardt J.R., Tony Stewart, Danica Patrick, or Jimmie Johnson. The company says 5 cards is equally likely to appear in any box of cereal. For a fan it takes 23 boxes to get all 5 drivers' cards-- should she be surprised? Design and carry out a simulation to help answer this question.
State: What is the probability that it will take 23 or more boxes to get a full set of 5 NASCAR collectible cards
Plan: We are going to assign the 5 drivers into five numbers: Jeff = 1, Dale = 2, Tony = 3, Danica = 4, Jimmie = 5. We'll use randInt(1,5) to simulate buying one box of cereal and looking at which card is inside. Because we want a full set of cards, we'll keep pressing Enter until we get all five of the labels from 1 to 5. Record the number of boxes that we had to open.
Do:
- REP 1: 3 5 2 1 5 2 3 5 4--> 9 boxes
- REP 2: 5 1 2 5 1 4 1 4 1 2 2 2 4 4 5 2--> 16 boxes
- REP 3: 5 5 5 2 4 1 2 1 5 3 --> 10 boxes
- REP 4: 4 3 5 3 5 1 1 1 5 3 1 5 4 5 2--> 15 boxes
- REP 5: 3 3 2 2 1 2 4 3 3 4 2 2 3 3 3 2 3 3 4 2 2 5--> 22 boxes
Conclude: We never had to buy more than 22 boxes to get the full set of cards in 50 repetitions of our simulation. SO our estimate of probability that it takes 23 or more boxes to get a full set is roughly 0. She should be surprised by how many boxes she had to buy.
Limits to the simulation: (5.1)
For the lottery example, does that mean it is inherently rigged?
- no because we just estimated the probability of getting two winners from the AP statistics class.
What about the cereal box simulation:
- It took our fan 23 boxes, although it seems very unlikely, it is possible she just was unlucky. The evidence suggests that the company's statement is incorrect.
Class example (5.1)
Samarth hits his first serve 55% of the time. We observe Samarth's serve 12 times and 5 land in. Run a simulation to estimate the probability that 5 of 12 land in.
- Let 01 to 55 to represent first serve lands in and 56 to 00 to represent a miss. Read a random digit table two digits at a time, starting at line 115, and record the number of times the first serve lands in. Continue reading from where left off, and repeat for a total of 5 times. The proportion of times 5 of the 12 serves land in is the estimated probability
Trials: 1 2 3 4 5
- 6/12, 7/12, 9/12, 8/12, 8/12
None of them reached 5/12, therefore the probability of that happening is 0.
