You roll two ordinary, 6-sided dice. What is the expectation of the sum of the two values showing? What is the expectation of the maximum of the two values showing?
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You roll two ordinary, 6-sided dice. What is the expectation of the sum of the
two values showing? What is the expectation of the maximum of the two values
showing?
Step by step
Solved in 3 steps
- There are 7 balls hidden in a box (3 blue and 4 green). If you first select a green ball and do not replace in the box, what is the probability that the second ball you select from the box will also be green? [Your answer can be written either as a fraction or decimal]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.6. AND Since there are two zeros on the board, let's determine the probability that you would be able to cover them on 2 consecutive turns. This means you'd have to roll a zero and then roll another zero. You can think of this as an AND probability. You need to roll 0 AND then you need to roll 0 again. Since the result of the first roll will have no influence on the result of the second roll, you can use the following rule: P(A and B) = P(A) * P(B) a) Determine P(0 and 0). b) Determine 1 - P(0 and 0). c) Write a sentence describing the meaning of 1 - P( Oand 0 ).
- 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…Problem 1. You are playing a version of the roulette game, where the pockets are from 0 to 10and even numbers are red and odd numbers are black (0 is green). You spin 3 times and add up the values you see. What is the probability th at you get a total of 17 given on the first spin you spin a 2? What about a 3? Solve by simulation and analytically.Two Dice were tossed. What is the probability that the total of two dice will be greater than 9, given that the first die is a 5? (Show your work)
- Imagine playing a number guessing game. A side is a number from 0 to Nhe's holding it, and the other side is trying to find that number by taking turns guessing. Number-holding side estimatehe has to offer one of the following three options in response to the party that did it:1-Your guess is correct, you found the number I kept (Game Over).2-Your estimate is wrong, but you are closer to the correct estimate than the previous estimate.3-the wrong estimate and the correct estimate are further away than the previous estimate.To find the estimated number in an environment where all the information is these, astrategy will be followed: Make a prediction (N/2) from the exact middle of N with 1 Begin:Find out the answer to your guess. [answer=answer_ogren (guess)]If the answer is equal to 1, the game is over, you can leave.If the answer is equal to 2, you are going in the right direction, keep the forecast direction;If you're heading for small numbers, the new N is now N/2.Make a guess…Assuming that you are only using one deck, and there are only four cards on the table. If you have a 10 and a 7 showing, the dealer has a 10 showing, calculate the probability of the dealer's other card being an 8, 9, or a face card.During the 17th century, gambler de Méré asked Blaise Pascal an explanation for his game losses. His question was solved by Pascal and Fermat, and yielded the foundations of the theory of probability. Gamblers used to bet on the event of getting at least one ace in four rolls of a dice. As a game variation, De Méré proposed to use two die and roll them 24 times with a bet on having at least one double ace. De Méré thought the games were equivalent but he lost consistently. Explain why? Write a simulation program to empirically determine which one dice/two die game is more likely to win.
- Let's revisit our first problem, where we want to set up a series of chess matches so we can rank six players in our class. As we did before, we will assume that everyone keeps their chess rating a private secret; however, when two players have a chess match, the person with the higher rating wins 100% of the time. But this time, we are only interested in identifying the BEST of these six players and the WORST of these six players. (We don't care about the relative ordering or ranking of the middle four players.) Your goal is to devise a comparison-based algorithm that is guaranteed to identify the player with the highest rating and the player with the lowest rating. Because you are very strong at Algorithm Design, you know how to do this in the most efficient way. Here are five statements. A. There exists an algorithm to solve this problem using 6 matches, but there does not exist an algorithm using only 5 matches. B. There exists an algorithm to solve this problem using 7 matches,…An Event In the context of the probability experiment just described. Consider the event in which the two numbers on the two dice add to 12. What is the size of this set? Probability of an event (1/2) What is the probability of the event just described? Write your answer as a fraction of integers. For example, if the answer were 17/20 then enter 17/20 with no spaces or other symbols. You may reduce the fraction by cancelling factors common to the numerator and denominator but you do not need to do this. Probability of an event (2/2) What is the probability that at least one 4 appears? Write your answer as a fraction of integers. For example, if the answer were 17/20 then enter 17/20 with no spaces or other symbols. You may reduce the fraction by cancelling factors common to the numerator and denominator but you do not need to do this.Casinos have devised different automated mechanical methods for shuffling the cards.One such method divides the deck into to seven piles by placing each card randomlyeither on the top or at the bottom of one pile (i.e. each card has 14 possible placesto choose from). After that, the piles are put together to form the shuffled deck.Is this a good method? Can a gambler utilize this information to his advantage?