Probability based on Geometric distribution.
Mr. Nerdly, the long-time AP Statistics teacher at Galton High, always assigns ten problems for homework. On day, he decides to make an unusual offer to his class. "My little cherubs," he says, "I have a proposition for you. In place of giving you the typical ten terrific textbook teasers, I would gladly allow probability to play a pivotal part in the process." Not quite sure what Mr. Nerdly has in mind, the students ask him to explain his proposal. "When class begins each day, I will select a student at random using my trusty calculator. Then, I will give the lucky student the opportunity to guess the day of the week on which one of my friends was born." (There is some snickering in the room as students imagine Mr. Nerdly's friends.) "If the chosen student guesses correctly, then I will assign only one homework problem that night. If, on the other hand, your representative gives the wrong day of the week, he or she will try to guess they say on which another Nerdly friend was born. This time, a correct answer will net you two homework problems. We will continue this little game until the chosen one's guess matches the day on which one of my acquaintances emerged from the womb. I will then assign you a number of homework questions equal to the number of guesses made by your chosen spokesperson. What you say?"
Before you make a decision about Mr. Nerdly's decision, why not try the birthday game from yourself. You might want to play several times before you draw any conclusions.
Mr. Nerdly's birthday challenge is an example of a geometric probability problem. For each of Mr. Nerdly's friends, the lucky student has a 1/7 chance of correctly guessing his/her day of birth. The trials (birthday guesses) are independent. The game continues until the first correct guess is made. In statistical language, we count the number of trials (birthday guesses) up to and including the first "success" (birthday match). If we let X=the number of guesses the student makes until he/she matches a Nerdly friend's day of birth, then X is a geometric random variable. We will return to this problem later in the unit.
1. What is the theoretical probability that Mr. Nerdly assigns 10 homework problems consequently of a randomly selected student playing the birthday game?
2. Find the theoretical probability that Mr. Nerdly assigns less than typical 10 homework problems as a result of a randomly selected student playing the birthday game?
3. Compute the probability that the number of homework problems Mr. Nerdly assigns as a result of playing the birthday game is within one standard deviation of the expected value for this game.