420sp22mid1qz
.pdf
School
Johns Hopkins University**We aren't endorsed by this school
Course
EN.553 550.420
Subject
Mathematics
Date
Dec 27, 2024
Pages
10
Uploaded by clarajeon126
553.420/620 Probability - Spring 2022Test #1Print your name here:Ethics statement:I agree to complete this examination without unauthorized assistance from any person, materi-als, or device. No internet, no cell phones, no texting, no human contact!Signature:X∼Bernoulli(p):P(X=x) =px(1-p)1-x,x= 0,1.X∼binomial(n, p):P(X=x) =(nx)px(1-p)n-x,x= 0,1,2, . . . , n.X∼hypergeometric:P(X=x) =(Mx)(N-Mn-x)(Nn),x= 0,1,2, . . . , n,x≤Mandn-x≤N-M.X∼Poisson(μ):P(X=x) =e-μ μxx!,x= 0,1,2,3, . . .X∼geometric(p):P(X=x) =p(1-p)x-1,x= 1,2,3, . . .X∼neg.binom.(r, p):P(X=x) =(x-1r-1)pr(1-p)x-r,x=r, r+ 1, r+ 2, . . .
1.Old MacDonald actually has a farm now. His farm has one very special chicken that layseggs.Assume that eggs laid are either satisfactory or unsatisfactory, and the probability ofa satisfactory egg is 0.35, independent from egg-to-egg.State the distribution (with properparameters) and support of each random variable below.No justification needed.(a)S15, the random number of eggs that are satisfactory in a batch of 15 eggs.(b)T3, the random number of eggs that the chicken needs to lay before seeing the thirdunsat-isfactoryegg.(c)U8, the random number ofunsatisfactoryeggs we obtain in a sample of 8 eggs from a groupof 10 satisfactory and 15 unsatisfactory eggs.
2.These are separate questions.(a) 6 chips numbered 1 through 6 are in a hat.We pull out the chips one by one withoutreplacement noting the number on each draw. Compute the probability we draw an even num-ber among the last two draws. For instance, (1,2,3,5,4,6),(2,4,1,3,6,5) and (4,3,6,2,4,1)belong to this event.(b) Consider all strings ofkletters (from 26 possible) with repetition allowed - each letterequally-likely. LetAkbe the event that ak-letter string is a palindrome. Recall that a palin-drome is a sequence of letters that reads the same both backwards and forwards, such as “level”,but palindromes can be nonsensical words, too, like “dzzd”. Determine which (if any) of thefollowing are true: (circle all that apply)P(A4) =P(A5),P(A4)> P(A5),P(A4)< P(A5).Justify your assertion.
3.The final exam for a certain math class is graded pass/fail. A randomly chosen student froma probability class has a 40% chance of knowing the material well. If she knows the materialwell, she has an 80% chance of passing the final. If she doesn’t know the material well, she hasa 40% chance of passing the final anyway. If the student passes, what is the probability thatshe knows the material?
4.Dan is playing the Pokemon Trading Card Game, and he uses the coin in the box todetermine who makes the first move. One side of the coin has a Pikachu on it, while the otherhas nothing.Let 0< p <1 be the probability the coin lands with Pikachu facing up. Danknows that the probability he must flip the coin exactly 3 times for the first Pikachu to appearis twice the probability he must flip the coin exactly 5 times for the first Pikachu to appear.Find the value ofp.
5.Maryland uses license plates that consist of 3 letters (A-Z), followed by 3 digits (0-9). Ifeach three-character arrangement letters and digits are equally likely, find the probability thata randomly created license plate has at least 1 palindrome.Recall:a palindrome is a sequence of characters that reads the same forwards as it doesbackwards, eg., ADA and 747 are both three-character palindromes.
6.Jake is dealing out 2 cards to each player at a poker table from a standard 52-card deck.Before dealing, he notices that one random card from the deck is missing, and he is too lazy tosearch for which one it is. Thus, he starts dealing out cards from the remaining 51-card deck.Find the probability that the first person to receive their two cards obtains a pocket pair, whichis two cards of the same rank, for instance, Ace♣, Ace♥.
7.To celebrate his 5 straight years on the Dean’s List, Carlos wants to put 5Xtattoos on thefinger(s) of his left hand - excluding his thumb. Each of these fingers can receive any or all oftheX’s. See the picture for some possibilities. Find the number of distinct ways Carlos can gettattooed in this manner.
8.A bin contains 7 basketballs, of which 5 are new (and 2 have been played with before).Suppose that 3 balls are selected randomly, played with, and after play are returned to the bin.Then, another 3 balls are selected for play a second time. What is the probability that theyare all new?Please simplify your fraction.Of course, assume all selections of 3 balls are equally-likely.
9.Gabe is arranging the textbooks he never used for his classes. He has 12 textbooks, of which3 of them are math textbooks. He’s moving out for the semester, so he puts all the books instorage boxes with maximum capacities of 3 books, 4 books, and 5 books. Gabe is in a hurry,so he fills up the boxes with books selected completely at random. Find the probability thatall 3 of the math textbooks are in the same box.Do not attempt to simplify your answer.