5. There are 10 people in a room. Your experiment is to obtain the birthdays of...

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5. There are 10 people in a room. Your experiment is to obtain the birthdays of these 10 people. [Assume that there are 365 possible birthdates for each person, so leap years are excluded.] (a) How many possible outcomes are possible in this experiment? [STOP! Do not enumerate!] (b) How many possible outcomes in which all 10 people have different birthdays? (c) Assuming that each of the 365 possible birthdates are equally likely (in reality may not be the case!), what is the probability that all 10 people have different birthdates? (d) What is the probability that at least two of them have identical birthdates? Is the value close to .50? (e) Generalize! Determine the minimum number of people you need in a room so that the probability that at least two of them have the same birthdates is at least .50. [Hint: Compute the probabilities when you have 15, 20, and 25 people. Then backtrack to get as close as possible to the probability of .50.] (f) Are you surprised with your answer in (e)? Did you intuitively expect that you needed more people? 5. There are 10 people in a room. Your experiment is to obtain the birthdays of these 10 people. [Assume that there are 365 possible birthdates for each person, so leap years are excluded.] (a) How many possible outcomes are possible in this experiment? [STOP! Do not enumerate!] (b) How many possible outcomes in which all 10 people have different birthdays? (c) Assuming that each of the 365 possible birthdates are equally likely (in reality may not be the case!), what is the probability that all 10 people have different birthdates? (d) What is the probability that at least two of them have identical birthdates? Is the value close to .50? (e) Generalize! Determine the minimum number of people you need in a room so that the probability that at least two of them have the same birthdates is at least .50. [Hint: Compute the probabilities when you have 15, 20, and 25 people. Then backtrack to get as close as possible to the probability of .50.] (f) Are you surprised with your answer in (e)? Did you intuitively expect that you needed more people?