HW1 - First problem set PDF

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First problem set...

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APMA 1650 HW1 Homework is due by 12pm (noon) in any one of the three dropoff boxes located in the first floor of 170 Hope street on the day listed on the course webpage. Show your work in deriving your solutions. Solutions must be written independently. Calculators and/or computers are allowed. Be sure to attach the homework 1 cover sheet or lose 5pts. If you worked with someone, please list them as collaborators on the coversheet. Recall, from Chapter 1 for a given data set {y1 , y2 , . . . yn } the sample mean is defined by 1X yi , y¯ = n i=1 n

while the sample variance is given by 1 X (yi − y¯)2 n−1 n

s2 =

i=1

1.[5pts] The United States Geological Survey on the magnitude of earthquakes greater than 6.0 in California. In 2008 there were 55 quakes of the following magnitude, 6.1 6.3 6.5 6.8 7.2

6.1 6.4 6.5 6.8 7.2

6.1 6.4 6.6 6.9 7.2

6.1 6.4 6.6 6.9 7.3

6.1 6.4 6.6 7.0 7.3

6.2 6.4 6.7 7.0 7.3

6.2 6.4 6.7 7.0 7.3

6.2 6.4 6.7 7.1 7.4

6.2 6.5 6.8 7.1 7.8

6.3 6.5 6.8 7.1 7.8

6.3 6.5 6.8 7.2 7.9

(a) Construct a relative frequency histogram for this data set. (b) What is the sample mean? (c) What is the sample variance? (d) What is the relative frequency of quakes within one standard deviation of the sample mean? (e) Based on this data how likely would you say it is to for California to experience an earthquake above 7.0? 2.[20pts] (Birthday problem) Consider a group of n people and assume that their birthdays are equally likely to fall on any day of the year. Ignore leap years so that the days of the year can be numbered from 1 to 365. As we did in class, we choose the sample space S to be the set of all possible sequences of n birtdays (i.e. a list of n numbers from 1 to 365).

(a) Let A be the event that someone in the group shares your birthday (assuming you are not part of the group). Find an exact formula for P (A). What is the smallest group size n such that P (A) > .5? (Use a calculator). Give a simple heuristic for why n must be greater then 365/2. (b) Let B be the event that two people have the same birthday. Find an exact formula for P (B). What is the smallest group size n such that P (B) > .5? 3.[10 pts] You forgot your password to your email account! You know that it must be one of your 16 passwords that you use for all your accounts (I don’t recommend this!). You decide to just start trying your passwords equally likely at random, never re-trying a password you’ve already tried. Unfortunately, after 5 failed attempts, your will be locked out of your account. What is the probability that you try the correct password before you get locked out? 4.[20pts] (a) How many unique permutations of the word ABRACADABRA are there? (b) In how many different ways can we arrange letters of the word ABRACADABRA so that C and D are not next to each other? 5. [25pts] Suppose you flip a fair coin 20 times. (a) [5pts] What is the probability of flipping exactly 5, 10, 12 Heads? (b) [10pts] What is the probability of flipping more Heads than Tails? (c) [10pts] What is the probability that you do not flip two Heads nor two Tails in a row? 6.[20pts] In poker, you are dealt a 5 cards. One-pair is the most common hand. After that, the next most common hands are two-pair and three-of-a-kind. Recall two-pair means that two cards have one rank, two cards have another rank, and the remaining card has a third rank. e.g. {2♥, 2♠, 5♥, 5♣, K♦} and three-of-a-kind means that three cards have one rank and the remaining two cards have two other ranks. e.g. {2♥, 2♠, 2♣, 5♣, K♦}. Calculate the probability of each type of hand. Which is more likely?...