Answer all the questions

b) What is the probability that Maria has fewer than & successes ? 3. A principal has 80 keys on her keychain to distribute to staff on the first day of school. Out of the 80 keys, 60 open classroom doors, 40 open the door to the teachers' lounge and 30 open both classroom doors and teachers lounge. Using this information, answer the following questions. Let C be the event that a randomly selected key opens a classroom and T be the event that a randomly selected key opens the teachers' lounge. a) What is P(C) b) What is P(T) c) What is P( C and T) d) What is P(C or T) e) Are the events, "a randomly selected key opens a classroom" and "a randomly selected key opens the teachers' lounge" mutually exclusive? Explain.1. A newly married couple wants to have three children. What are the different combinations of children they could have? a) Write the sample space, S, for these outcomes. b) What is the probability of having all boys? C ) Are the probabilities obtained from this example theoretical or empirical? Explain. 2. The probability that Maria succeeds at any given free-throw is 75%. She was curious how many free-throws she can expect to succeed in a sample of 10 free-throws. She simulated 25 samples of 10 free-throws where each free-throw had a 0.75 probability of being a success Maria counted how many free-throws were successes in each simulated sample. Here are her results: a) Is it unusual that Maria gets 6 free-throws out of ten in this simulation? Explain your reasoning. ........ CO.... ..... ..... 5 6 7 8 9 10 # of free-throw successes4. A survey was done of commuters in three major cities about how they primarily got to work. The results are shown in the frequency table below. Car Train Walk Total New York 10 50 20 80 Los Angeles 36 24 10 70 Chicago 16 28 6 50 Total 62 102 36 200 Find the following probabilities if one commuter is selected at random. a.) P(NY) b.) P(NY and Car) c.) P(NY or Car) d.) P(Train | NY) e.) P(Train | LA) f.) P(Train | Chicago) g.) In which city is a commuter most likely to take the train as a method of transportation? Explain your