Ground Rules: Some collaboration is acceptable provided that you document it above. However, each person must submit a unique writeup. Collaboration may include sharing ideas and cross-checking answers for reasonableness. Collaboration shall not include copying work, copying or linking TI-83 programs, using a tutor to lead you step-by-step through a solution, or any other "shortcuts" that deny you the opportunity to learn something from this assignment. If in doubt, send e-mail or leave voice mail at (703) 599-6624. By your signature above, you are certifying that the work in this assignment is essentially your own, although you may have shared a few ideas and may have compared answers with other students for reasonableness. Other ground rules include some do’s and don’ts:

Do: give fractional answers where possible.
Do: give final decimal answers in addition to fractional answers.
Do: round correctly in final answer, to 3 decimal places or 3 significant figures, whichever gives greater precision.
Don’t: write calculator notation (e.g., don’t say 3.45E-4, say 0.000345).
Don’t: truncate or round intermediate results (use STO key to store to a working variable).

Scoring Criteria: Each problem is worth 15 points. For my information only, please indicate (in the blank above) how long you spent working on this assignment; this figure will not affect your score in any way. Work is crucial: a correct answer with insufficient work will earn zero points, but a wrong answer with clear, neat, organized work may earn nearly the maximum number of points. If your handwriting is terrible, please rewrite your scratch work slowly enough so that it is linearly organized and legible. Spelling counts for a few points, too.

From an ordinary deck of 52 playing cards (no jokers), we remove all the kings, jacks, and queens. We are left with 40 cards, consisting of the ace, deuce, three, four, ..., nine, and ten of each of four suits. Our experiment (one repetition, that is) will consist of 5 trials. In each trial we shuffle the 40 cards, draw a card, record its value (A, 2, 3, 4, 5, 6, 7, 8, 9, or 10), and replace the card in the deck. Thus, even though we are "drawing 5," each draw is always made from a shuffled 40-card deck.

(a) Are the trials independent events? ________ Why or why not? _____________________________________

(b) What is the exact probability of getting at least one ace in 5 draws? Show your work.

(c) On a separate sheet of paper (or on the back side of p.3), design a simulation to verify empirically that your answer to part (b) is reasonable. Perform the experiment (and by that I mean 5 draws) at least 20 times. Of course, more repetitions will probably give better accuracy and may help you decide if your answer to part (b) is correct, but you can earn full credit with only 20 repetitions. Be sure to indicate how you are assigning card values to random digits. You may use either the random digit table (starting at row _______ ) or a TI-83 calculator program, or both. If you use the random digit table, mark it up clearly to indicate your work. If you write a calculator program (which is normally not permitted for credit on the AP exam), be sure include your output and an annotated program listing that briefly describes the purpose of each significant line. (Calculator programs without a listing, or with a listing that is not sufficiently annotated, will earn zero points.) Your final answer should be an estimate for the probability you computed in part (b).

(d) Approximately how many repetitions would be required in your simulation to achieve a 95% confidence estimate for part (b) that would be accurate within 0.01? (We don’t know how to answer this yet, so just make a guess.) Answer: about ____________ repetitions of 5 trials each.

(a) Imagine taking the 8 letters in the word TANGENTS and making each one a different color. In how many ways can a left-to-right sequence of 5 of these letters be created? Note: We would count STATE twice because the two Ts are distinguishable. Patterns need not spell an actual word.

(b) Now imagine that the letters are all the same color. How many unique 5-letter patterns are possible? Hint: It is very helpful to consider cases. Try counting all the patterns that have 1 T and no Ns, then no Ts and 1 N, then 2 Ts and no Ns, 2 Ts and 1 N, etc. Note that in part (b) we would count STATE only once, because the two Ts are indistinguishable.

(c) If the 8 letters in TANGENTS are shaken in a bag and then selected, in order, by a special machine that ensures that there is no bias in the probabilities of drawing letters, what is the probability of spelling the word GATES? The letters G, A, T, E, and S must be drawn in that order, without replacement.

(d) Repeat problem (c), except this time, compute the probability for the word AGENT.

(e) As the machine plucks letters out of the bag, are these selections independent events? ________ Why or why not? _____________________________________

(f) Your answers to parts (c) and (d) differ by a factor of ____________ . Use part (a) to explain why. (Yes, I know you can explain why simply by comparing parts (c) and (d) numerically, but give an explanation that uses part (a).)