# Randomness, Probability, and Simulation

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Randomness, Probability, and Simulation
Section 5.1 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore Lesson 6.1.1

Activity! The “1 in 6 wins” Game
As a special promotion for its 20-ounce bottles of soda, a soft drink company printed a message on the inside of each bottle cap. Some of the caps said, “please try again!” while others said “you’re a winner!” the company advertised the promotion with the slogan “1 in 6 wins a prize.” Seven friends each buy one 20-ounce bottle at a local convenience store. The store clerk is surprised when three of them win a prize. Is this group of friends just lucky, or is the company’s 1-in-6 claim inaccurate?

Activity! For now, lets assume the company is telling the truth, and that every 20-ounce bottle of soda it fills has a 1-in-6 chance of getting a cap that says “you’re a winner” We can model the status of an individual bottle with a six-sided die: Let 1 to 5 represent “please try again!” 6 represent “you’re a winner”

Activity! Roll your die seven times to imitate the process of the seven friends buying their sodas. How many of them won a prize? Repeat step 1 four more times. In your five repetitions of this simulation, how many times did three or more of the group win a prize? Combine results with your classmates. What percent of the time did friends come away with three or more prizes, just by chance? Based on your answer in step 3, does it seem plausible that the company is telling the truth, but that the seven friends just got lucky?

Objectives Idea of probability Myths about Randomness Simulations
Law of large numbers Probability Myths about Randomness Runs “hot hand” “law of averages” Simulations Preforming simulations State Plan Do Conclude

Idea of Probability The big fact emerges when we look at random sampling or random assignment closely: chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run. This remarkable fact is the basis of probability!

Law of Large Numbers The fact that the proportion of heads in many tosses eventually closes in on .5 is guaranteed by the law of large numbers. With more and more times the situation is being conducted, the pattern emerges into what's known as the probability.

Probability 0….……………… (.5)………………….1 0% ………………(50%)……………..100%
The probability of an outcome of a chance process is a number between 0 and 1 that describes the proportion of times that outcome would occur in a very long series of repetitions. 0….……………… (.5)………………….1 0% ………………(50%)……………..100% Outcomes that never occur have a probability of 0 An outcome that happens on every repetition has probability of 1

Check for Understanding
According to the books of odds, the probability that a randomly selected U.S. adult usually eats breakfast is 0.61 Explain what probability 0.61 means in this setting Why doesn’t the probability say that if 100 U.S adults are chosen at random, exactly 61 of them usually eat breakfast?

Check for Understanding
Probability is a measure of how likely an outcome is to occur. Match one of the probabilities that follow with each statement. Be prepared to defend your answer This outcome is impossible. It can never occur. This outcome is certain. It will occur on every trial. This outcome is very unlikely, but it will occur once in a while in a long sequence of trials This outcome will occur more often than not.

The idea of probability seems straightforward. It answers the question “what would happen if we did this many times?” both the behavior of random phenomena and the idea of probability are a bit subtle. We meet chance behavior constantly, and psychologists tell us that we deal with it poorly…

Unfortunately, it is out intuition about randomness that tries to mislead us into predicting the outcome in the short run… Toss a coin six times and record heads (H) or tails (T) on each toss. Which of the following outcomes is more probable? HTHTTH TTTHHH

BOTH! Almost everyone says that HTHTTH is more probable because TTTHHH does not “look random.” In fact both are equally likely. The coin has no memory, it doesn’t know what past outcomes were, and it cant try to create a balanced sequence.

Myth: Runs More examples:
The outcome TTHHH in tossing six coins looks unusual because of the runs of 3 straight tails and 3 straight heads. Runs seem “not random” to out intuition but are quite common. More examples: “she’s on fire!” (basketball player makes several shots) “Law of Averages” Roulette tables in casinos. “red is due!” from the display board

Simulations and Preforming Them
Simulation: The imitation of chance behavior, based on a model that accurately reflects the situation is called a simulation

Simulations and Preforming Them
Statistics Problems Demand Consistency! State: What is the question of interest about come chance process? Plan: Describe how to use a chance device to imitate one repetition of the process. Explain clearly how to identify the outcomes of the chance process and what variables to measure. Do: Perform many repetitions of the simulation Conclude: Use the results of your simulation to answer the question of interest.

Example: 6 in 1 game! State What’s the probability that three of seven 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. 6 = wins a prize 1 to 5 = no prize Roll the dice seven times, once for each person Record whether 3 or more people win a prize (yes or no) Do Each student perform 5 repetitions Conclude Out of 125 total repetitions of the simulation, there were 15 times when three or more of the seven people won a prize. So our estimate of the probability is 15/125, or about 12%

Warning: phrasing your Conclusions
When students make conclusions, they often lose credit for suggesting that a claim is definitely true or that the evidence proves that a claim is incorrect. A better response would be to say that there is sufficient evidence (or there isn’t sufficient evidence) to support a particular claim.

Objectives Idea of probability Myths about Randomness Simulations
Law of large numbers Probability Myths about Randomness Runs “hot hand” “law of averages” Simulations Preforming simulations State Plan Do Conclude

Homework Worksheet