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3. A computer chooses 100 independent random numbers, each uniformly from the set of integers between...
In a certain lottery, players pick 5 numbers (without replace) from the numbers 1-50 and an...
In a certain lottery, players pick 5 numbers (without replace) from the numbers 1-50 and an additional number (possibly repeated from the first set) from the numbers 1-30. A set of 6 numbers with these restrictions is then chosen uniformly at random. The player wins based on how many of their numbers matched the randomly chosen number. (a) What is the probability that the player chooses all 6 numbers correctly? (b) What is the probability that the player chooses exactly...
A letter is chosen uniformly at random from {A, B, . . . , Z}. If...
A letter is chosen uniformly at random from {A, B, . . . , Z}. If that letter is one of the vowels (i.e. A, E, I, O or U) then a second letter is chosen uniformly at random from {A, B, . . . , Z}. Let L be the number of letters chosen and let V be the number of vowels chosen. (i) What is the expected value of L? (ii) What is the expected value of V?...
2. Suppose an integer is chosen at random from the set S of the first 2510 positive integers that is, from the set S- [...
2. Suppose an integer is chosen at random from the set S of the first 2510 positive integers that is, from the set S- [1,2,3,...,2510). Let A be the event that the number chosen is a multiple of 47. Let B be the event that the number chosen is a multiple of 23. (a) Determine with reason whether the events A and B are mutually exclusive. (b) Determine with reason whether the events A and B are independent (c) Determine...
Counting and Pigeonhole Principle (a). A set of four different integers is chosen at random between...
Counting and Pigeonhole Principle (a). A set of four different integers is chosen at random between 1 and 200 (inclusive). How many different outcomes are possible? (b). How many different integers between 1 and 200 (inclusive) must be chosen to be sure that at least 3 of them are even? (c). How many different integers between 1 and 200 (inclusive) must be chosen to be sure that at least 2 of them add up to 20? (d). How many different...
Alexa1 and Zoltan2 play the following game: AZ-game: Step 1: Alexa chooses a uniformly random element...
Alexa1 and Zoltan2 play the following game: AZ-game: Step 1: Alexa chooses a uniformly random element from the set {1,2,3}. Let a denote the element that Alexa chooses. Step 2: Zoltan chooses a uniformly random element from the set {1, 2, 3}. Let z denote the element that Zoltan chooses. Step 3: Using one of the three strategies mentioned below, Alexa chooses an element from the set {1, 2, 3} \ {a}. Let a′ denote the element that Alexa chooses....
A hacker has programmed their computer to generate, uniformly at random, an eight-character password, with each...
A hacker has programmed their computer to generate, uniformly at random, an eight-character password, with each character being either one of 26 lower-case letters (a-z), one of 26 upper-case letters (A-Z) or one of 10 integers (0-9). The hacker wants to infiltrate a website that has 2 million users. Assume, for simplicity, that each user is required to use a unique password. i. What is the expected number of attempts before the hacker successfully generates a user password? ii. What...
A program for generating random numbers on a computer is to be tested. The program is...
A program for generating random numbers on a computer is to be tested. The program is instructed to generate 100 single-digit integers between 0 and 9. The frequencies of the observed integers were as follows. At the 0.05 level of significance, is there sufficient reason to believe that the integers are not being generated uniformly? Integer 0 1 2 3 4 5 6 7 8 9 Frequency 8 9 7 8 11 12 6 12 13 14 (a) Find the...
4.26 In a lottery game, three winning numbers are chosen uniformly at random from (1, ,100), samp...
4.26 In a lottery game, three winning numbers are chosen uniformly at random from (1, ,100), sampling without replacement. Lottery tickets cost $1 and allow a player to pick three numbers. If a player matches the three winning numbers they win the jackpot prize of $1,000. For matching exactly two numbers, they win $15. For matching exactly one number they win $3 d) Hoppe shows that the probability that a single parlayed ticket will ulti- mately win the jackpot is...
Assume that you select two 4-digit numbers at random from the set of consecutive integers from...
Assume that you select two 4-digit numbers at random from the set of consecutive integers from 0000 through 9999. The selections are made with replacement and are independent of one another. If the two numbers are the same, you make $650. If they are different, you lose $4. A random variable XX is defined as your gain or loss.
In the game “cobbler’s purse,” a player chooses a number in the set S = {2,...
In the game “cobbler’s purse,” a player chooses a number in the set S = {2, 3, . . . , 12}. and then rolls a pair of dice. The player wins if the sum of the results of the two dice is equal to the number chosen. a) Assuming that the player chooses their number from the set S, each with equal likelihood, what is the probability that they will win? (b) Assuming that the player chooses their number...
2. Suppose an integer is chosen at random from the set S of the first 2510 positive integers that is, from the set S- [1,2,3,...,2510). Let A be the event that the number chosen is a multiple of 47. Let B be the event that the number chosen is a multiple of 23. (a) Determine with reason whether the events A and B are mutually exclusive. (b) Determine with reason whether the events A and B are independent (c) Determine...
Counting and Pigeonhole Principle (a). A set of four different integers is chosen at random between 1 and 200 (inclusive). How many different outcomes are possible? (b). How many different integers between 1 and 200 (inclusive) must be chosen to be sure that at least 3 of them are even? (c). How many different integers between 1 and 200 (inclusive) must be chosen to be sure that at least 2 of them add up to 20? (d). How many different...
4.26 In a lottery game, three winning numbers are chosen uniformly at random from (1, ,100), sampling without replacement. Lottery tickets cost $1 and allow a player to pick three numbers. If a player matches the three winning numbers they win the jackpot prize of $1,000. For matching exactly two numbers, they win $15. For matching exactly one number they win $3 d) Hoppe shows that the probability that a single parlayed ticket will ulti- mately win the jackpot is...
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