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Explain what is meant by the long-run relative frequency definition of probability.

A teacher announces a pop quiz for which the student is completely unprepared. The quiz consists of 100 true-false questions. The student has no choice but to guess the answer randomly for all 100 questions. a. Simulate taking this quiz by random guessing. Number a sheet of paper 1 to 100 to represent the 100 questions. Write a T (true) or F (false) for each question, by predicting what you think would happen if you repeatedly flipped a coin and let a tail represent a T guess and a head represent an F guess. (Don't actually flip a coin; merely write down what you think a random series of guesses would look like.) b. How many questions would you expect to answer correctly simply by guessing? c. The table shows the 100 correct answers. The answers should be read across rows. How many questions did you answer correctly? d. The preceding answers were actually randomly generated by an app. What percentage were true, and what percentage would you expect? Why are they not necessarily identical? e. Are there groups of answers within the sequence of 100 answers that appear nonrandom? For instance, what is the longest run of Ts or Fs? By comparison, which is the longest run of Ts or Fs within your sequence of 100 answers? (There is a tendency in guessing what randomness looks like to identify too few long runs in which the same outcome occurs several times in a row.)

Your teacher gives a true-false pop quiz with 10 questions. a. Show that the number of possible outcomes for the sample space of possible sequences of 10 answers is \(1024 .\) b. What is the complement of the event of getting at least one of the questions wrong? c. With random guessing, show that the probability of getting at least one question wrong is approximately \(0.999 .\)

Big loser in Lotto Example 10 showed that the probability of having the winning ticket in Lotto South was 0.00000007 . Find the probability of holding a ticket that has zero winning numbers out of the 6 numbers selected (without replacement) for the winning ticket out of the 49 possible numbers.

In criminal trials (e.g., murder, robbery, driving while impaired, etc.) in the United States, it must be proven that a defendant is guilty beyond a reasonable doubt. This can be thought of as a very strong unwillingness to convict defendants who are actually innocent. In civil trials (e.g., breach of contract, divorce hearings for alimony, etc.), it must only be proven by a preponderance of the evidence that a defendant is guilty. This makes it easier to prove a defendant guilty in a civil case than in a murder case. In a high- profile pair of cases in the mid 1990 s, O. J. Simpson was found to be not guilty of murder in a criminal case against him. Shortly thereafter, however, he was found guilty in a civil case and ordered to pay damages to the families of the victims. a. In a criminal trial by jury, suppose the probability the defendant is convicted, given guilt, is \(0.95,\) and the probability the defendant is acquitled, given innocence, is 0.95 . Suppose that \(90 \%\) of all defendants truly are guilty. Given that a defendant is convicted, find the probability he or she was actually innocent. Draw a tree diagram or construct a contingency table to help you answer. b. Repeat part a, but under the assumption that \(50 \%\) of all defendants truly are guilty. c. In a civil trial, suppose the probability the defendant is convicted, given guilt is 0.99 , and the probability the defendant is acquitted, given innocence, is \(0.75 .\) Suppose that \(90 \%\) of all defendants truly are guilty. Given that a defendant is convicted, find the probability he or she was actually innocent. Draw a tree diagram or construct a contingency table to help you answer.