Geometric Probability

[Stunt101]

Can someone help me with this problem and check my answer..
(I know its a long read but the questions are not hard)

Mr. Nedrdly, the long time ap statistics teacher at galton high, always assigns ten problems for homework. One day, he decides to make an unusal offer to his class. "my litter cherubs", he says, "i have a proposition for you. Instead 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 waht mr. nerdlyu 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 olpportunity to guess the day of the week on which one of my many friends was born." "if the chosen student guesses correctly, then i wsill 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 the day 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 matchs the day on whihc one of my acquatintances 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 say you?"

Mr. Nerdly's birthday challenge is san example of a geometric probability problem. For each of Mr. nerdly's friends, the lucky student has a 1/7 chance of correctly guesing his/her day of birth. The trials (birthday guesses) are independent. The game contiunes 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 (birdthday match. If we let X=the number of guesses the stundet makes until he/she matches a nerdly friend's day of birth, the X is a geometric random variable.

Questions:
1.whis the theoretical probabilty that Mr. Nerdly assigsn 10 homework problems as a result of a randomly selected student playing the birthday game? My answer:.0357
2.Find the theoretical probability that Mr. Nerdly assigns less than the typical 10 homework problems as a result of a randomly selected student playing the birthday game? My answer:.7861
3.Calculate the probability that the number of homework problems Mr. Nerdly assigns as a result of playing the birthday game is withing one standard deviation of the expected value for this game? I dont know how to do this one...

Thanks for any help

[F.A.P]

Questions:
1.whis the theoretical probabilty that Mr. Nerdly assigsn 10 homework problems as a result of a randomly selected student playing the birthday game? My answer:.0357
2.Find the theoretical probability that Mr. Nerdly assigns less than the typical 10 homework problems as a result of a randomly selected student playing the birthday game? My answer:.7861
3.Calculate the probability that the number of homework problems Mr. Nerdly assigns as a result of playing the birthday game is withing one standard deviation of the expected value for this game? I dont know how to do this one...

The number of problems N is geometric or more preferably denoted FirstSuccess(p) with p = 1/7. It has pmf

P(N = k) = p*(1 - p)^(k-1)

1. P(N = 10) = (1/7)*(6/7)^(9) =~ 0.0357

2. P(N < 10) = P(N = 1) + P(N = 2) +...+ P(N = 9) =~ 0.750

3. Expected value and variance is found by summation to be

E(N) = 1/p = 7, Var(N) = (1 - p)/p^2 = 42

Now the probability to be within 1 stdev from the expectation value is

P(7-sqrt(42) < N < 7 + sqrt(42)) = P(0 < N < 14) = P(N < 14) =

= P(N = 1) + P(N = 2) +...+ P(N = 13) =~ 0.865