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

Your teacher gives a truefalse 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 \(0.999 .\)

A couple plans on having four children. The father notes that the sample space for the number of girls the couple can have is \(0,1,2,3,\) and \(4 .\) He goes on to say that since there are five outcomes in the sample space, and since each child is equally likely to be a boy or girl, all five outcomes must be equally likely. Therefore, the probability of all four children being girls is \(1 / 5 .\) Explain the flaw in his reasoning.

A standard card deck has 52 cards consisting of 26 black and 26 red cards. Three cards are dealt from a shuffled deck, without replacement. a. True or false: The probability of being dealt three black cards is \((1 / 2) \times(1 / 2) \times(1 / 2)=1 / 8 .\) If true, explain why. If false, show how to get the correct probability. b. Let \(A=\) first card red and \(B=\) second card red. Are A and B independent? Explain why or why not. c. Answer parts a and b if each card is replaced in the deck after being dealt.

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.