A coin is flipped 8 times in a row (assume all outcomes are equally likely).For each of the following questions, you should write your answer as an expression. Do not give thefinal numeric value. For example, you should write C(4,2)/24 instead of 0.375.Q1.1 What is the probability that it lands on heads exactly four times?

Answered: Image /qna-images/answer/a0d49642-fb42-4514-9453-0df37aa8a17d.jpg

Answered: A coin is flipped 8 times in a row…

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter17: Markov Chains

Section: Chapter Questions

Problem 12RP

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There are 13 ranks and 4 suits in a standard deck of cards. Each card has exactly one rank and one suit, making 13.4 52 cards. a In poker, you randomly draw a hand of 5 cards (and play all five cards). What is the probability of drawing a flush, i.e. five cards of the same suit? b In the video game Balatro, you draw 8 cards but still play only 5 cards. What is the probability of being able to play a flush? Hint: consider what could be the best/most efficient way to count this to avoid double counting. Rephrasing the question might help.
EXERCISE - You go to a market to pick out some gourds to decorate your house for Halloween. The market has a special October deal where you get 3 randomly chosen gourdss for $10. Each gourd can either be an orange pumpkin, a green pumpkin or a squash. The probabilities for picking each one of them are: P(orange pumpkin) = 0.6, P(green pumpkin) = 0.3, and P(squash) = 0.1. A: What is the probability that the first gourd is an orange pumpkin, the second is a green pumpkin, and the third is a squash? B: What is the probability that you get all 3 as orange pumpkins? C: What is the probability that you get no squashes? D: What is the probability that you get at least one orange pumpkin? (ANSWERE USING PYTHON)
Q1. Let’s play a dice game with a pair of dice following these rules:1. At the beginning, you throw a pair of dice. If the two numbers add up to 5, 6, 7, 8, or 9, thegame immediately stops.2. If your first throw does not meet those 5 totals, you would continue until you get either 11 or12.Get 1000 simulations of this paired dice game. What is the average number of dice throw per game?You can use the sample() function to simulate the dice.
he Dice game of “Pig” can be played with the following rules. Roll two six-sided dice. Add the face values together. Choose whether to roll the dice again or pass the dice to your opponent. If you pass, then you get to bank any points earned on your turn. Those points become permanent. If you roll again, then add your result to your previous score, but you run the risk of losing all points earned since your opponent had rolled. Continue to roll as much as you want. However, once a “1” comes up on either die, your score is reduced to 0, leaving you only with points that you have previously "banked." Furthermore, you must pass the dice to your opponent. The first person to 100 points is the winner. When a player rolls two dice, the possible outcomes are as follows: Die #2 Roll Roll a 1 Roll a 2 Roll a 3 Roll a 4 Roll a 5 Roll a 6 Die #1 Roll Roll a 1 0 0 0 0 0 0 Roll a 2 0 4 5 6 7 8 Roll a 3 0 5 6 7 8 9 Roll a 4 0 6 7 8 9 10 Roll a 5 0 7 8 9 10 11 Roll a 6 0 8 9 10…
Suppose a person can buy a chocolate bar from the vending machine for $1 each. Inside every chocolate bar is a coupon. A person can redeem 3 coupons for one chocolate bar from the machine. This means that once a person has started buying chocolate bars from the machine, he/she always has some coupons. A person would like to know how many chocolate bars can be bought, if a person starts with N dollars and always redeem coupons, if he/she has enough for an additional chocolate bar. For example: With 3 dollars a person could buy 4 chocolate bars after purchasing 3 bars giving him/her 3 coupons and then redeeming the 3 coupons for one bar. This would leave him/her with one extra coupon. Thus, will have 4 chocolate bars and still have one coupon leftover. For 11 dollars, a person can have 16 chocolate bars and still have one coupon leftover. For 12 dollars, a person can have 17 chocolate bars and have two coupons leftover. Write a complete Python program that prompts a buyer to input…
In the Dice Roll game, the player begins with a score of 1000. The player is prompted for the numberof points to risk and a second prompt asks the player to choose either high or low. The player rollstwo dice and the outcome is compared to the player’s choice of high or low. If the dice total is between2 and 6 inclusive, then it is considered “low”. A total between 8 and 12 inclusive is “high”. A total of7 is neither high nor low, and the player loses the points at risk. If the player had called correctly, thepoints at risk are doubled and added to the total points. For a wrong call, the player loses the points atrisk. Create a DiceRollGame application that uses a DRPlayer object based on this specification. TheDRPlayer object should have two Die member variables that represent the dice. The Die class shoulduse a random number generator to determine the outcome in a roll() method. Application outputshould look similar to:
In the Dice Roll game, the player begins with a score of 1000. The player is prompted for the numberof points to risk and a second prompt asks the player to choose either high or low. The player rollstwo dice and the outcome is compared to the player’s choice of high or low. If the dice total is between2 and 6 inclusive, then it is considered “low”. A total between 8 and 12 inclusive is “high”. A total of7 is neither high nor low, and the player loses the points at risk. If the player had called correctly, thepoints at risk are doubled and added to the total points. For a wrong call, the player loses the points atrisk. Create a DiceRollGame application that uses a DRPlayer object based on this specification. TheDRPlayer object should have two Die member variables that represent the dice. The Die class shoulduse a random number generator to determine the outcome in a roll() method. Application outputshould look similar to:
Interest on a credit card’s unpaid balance is calculated using the average daily balance. Suppose that netBalance is the balance shown in the bill, payment is the payment made, d1 is the number of days in the billing cycle, and d2 is the number of days payment is made before billing cycle. Then, the average daily balance is: averageDailyBalance =netBalance x d1-payment x d2d1 If the interest rate per month is, say, 0.0152, then the interest on the unpaid balance is: Interest= averageDailyBalance * 0.0152 Write a program using c++ compiler that accepts as inputnetBalance, payment, d1,d2, and interest rate per month. The program outputs the interest. Format your output to two decimal places.
Dynamic Programming: Two coins X and YFlip coin X: gain 1 point for the head, and lose 1 point for the tailFlip coin Y: gain 3 points for the head, lose 3 points for the tail. After n round, if the final score is positive, you win. Otherwise, you lose. The score does not matter. Pr(s) be the probability of winning; r is the round; s is the score.P0(s) = 1 for s >= 1; P0(s) = 0 if s <= 0. a. Explain why P1(s) = 0 when s <= -3; P1(s) = 0.5 when -2 <= s <= 1; P1(s) = 1 for s >= 2. b. Explain why you must select X if s is 2 or 3, and you must select Y if s is −2 or −1. b. Determine P2(s) for each possible value of s and explain. (each probability will be one of 0, ¼, 1/2, ¾, 1) c. Fine a recurrence relation for Pr(s) with form Pr(s) = max[(?+?)/2, (?+?)/2]. Explain. d. Use Dynamic Programming to compute probability P100(0) (win the game if it has 100 rounds).
Write a program to ask user to enter number of codes. Assign a letter for each code and probabilities. Please make sure the total probabilities should be equal to 100%. Create a Huffman code for each letter.
Bowling involves 10 frames. Each frame starts with 10 pins. The bowler has two throws to knock all 10 pins down. The total score is the sum of pins knocked down, with some special rules. For the first 9 frames: If all 10 pins are knocked down on a frame's first throw (a "strike"), that frame's score is the previous frame plus 10 plus the next two throws. (No second throw is taken). If all 10 pins are knocked down after a frame's second throw (a "spare"), that frame's score is the previous frame plus 10 plus the next throw. In the 10th frame, if the bowler's first throw is a strike, or the first two throws yields a spare, the bowler gets a third throw. The 10th frame's score is the previous frame's score plus the pins knocked down in the 10th frame's two or three throws. Given integers represents all throws for a game, output on one line each frame's score followed by a space (and end with a newline). Note that the number of throws may be as few as 11 (strikes in first 9 frames,…
Bowling involves 10 frames. Each frame starts with 10 pins. The bowler has two throws to knock all 10 pins down. The total score is the sum of pins knocked down, with some special rules. For the first 9 frames: If all 10 pins are knocked down on a frame's first throw (a "strike"), that frame's score is the previous frame plus 10 plus the next two throws. (No second throw is taken). If all 10 pins are knocked down after a frame's second throw (a "spare"), that frame's score is the previous frame plus 10 plus the next throw. In the 10th frame, if the bowler's first throw is a strike, or the first two throws yields a spare, the bowler gets a third throw. The 10th frame's score is the previous frame's score plus the pins knocked down in the 10th frame's two or three throws. Given integers represents all throws for a game, output on one line each frame's score followed by a space (and end with a newline). Note that the number of throws may be as few as 11 (strikes in first 9 frames,…

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