6. A and B are playing a short game of ping pong where A serves 3...

6. A and B are playing a short game of ping pong where A serves 3 times and B also serves 3 times. If after these six points one of them is ahead the game ends, otherwise they go into a second phase. Suppose that A wins 70% of the points when they serve and 40% of the points when B serves.

Let’s look at the first phase.

a) (3 pts) Find the probability that A or B wins 0, 1, 2, or 3 points when they serve (give the answers separately, so P(A wins 0 points when A serves)= , ...).

b) (4 pts) Find the probability that A scores a total of 4 or more points (so wins in the first phase).

c) (2 pts) Find the probability that A scores 3 points in total (so there is a tie in the first phase).

Now let’s look at cases where the game moves on to the second phase. In this phase there are multiple rounds; in each round each player serves once. They win if they win both points; otherwise it goes to another round. Play continues until someone wins.

d) (4 pts) Find the probability that A wins if it goes to the second phase.

e) (2 pts) Find the probability that A wins (in either the first or second phase).

f) Extra credit (3 pts): find the expected number of points played.

Solutions

Expert Solution

milcah answered 9 months ago

Next > < Previous

Related Solutions

You are playing a version of the roulette game, where the pockets are from 0 to...

You are playing a version of the roulette game, where the pockets are from 0 to 10 and 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 that you get a total of 15 given on the first spin you spin a 2? What about a 3? Solve by simulation and analytically.

Suppose you’re playing a game where you consecutively roll 3 dice. After each roll you may...

Suppose you’re playing a game where you consecutively roll 3 dice. After each roll you may choose to either roll the next dice or sacrifice one die to reroll any number of the previous dice. If you get a number greater than 5 you win, but if you roll doubles or a number less than 6 you lose. Considering each roll a separate state what is the approximate branching factor? Justify. Draw the full state space considering only the number...

Alex and Owen are playing a game of simultaneous moves where they are deciding which concert...

Alex and Owen are playing a game of simultaneous moves where they are deciding which concert they will attend on Saturday night. Two musicians are performing that night locally, Artist 1 and Artist 2. Alex prefers to go to Artist 1 with Owen rather than Artist 2 with Owen, and he prefers going to Artist 2 with Owen rather than going to Artist 2 alone. Moreover, he prefers going to Artist 1 alone rather than going to Artist 2 with...

Peggy and Marcy are playing an ultimatum game where Peggy is given $500 and asked to...

Peggy and Marcy are playing an ultimatum game where Peggy is given $500 and asked to propose a way of splitting it with Marcy. When Marcy learns Peggy’s proposal, Marcy chooses whether to accept or reject the split. If Marcy accepts the split, both players receive the money according to Peggy’s split proposal. If Marcy rejects the split, both players receive nothing. This game will be played only once, so Peggy does not have to worry about reciprocity when making...

6. You are playing a card game with a friend. You are using a new deck...

6. You are playing a card game with a friend. You are using a new deck of 52 playing cards and you’d like to calculate some probabilities to improve your game. (Remember, the total number of cards decreases by 1 every time you draw a card!) a. What is the probability of drawing three queen cards in a row? b. What is the probability of drawing all four aces in a row? c. What is the probability of drawing the...

6. You are playing a card game with a friend. You are using a new deck...

Consider the following game, where p is the first player's probability of playing Up and q...

Consider the following game, where p is the first player's probability of playing Up and q is the second player's strategy of playing Left. Left (q) Right (1-q) Up (p) (2,6) (0,0) Down (1-p) (0,0) (6,2) What are the pure-strategy Nash equilibria? Group of answer choices (p,q)=(1,1) (p,q)=(1,0) (p,q)=(0,1) (p,q)=(0,0) Compute the probabilities for the mixed-strategy Nash equilibrium. Show your work step by step

A variance of Monty Hall Problem. Let's say I am playing a game where there are...

A variance of Monty Hall Problem. Let's say I am playing a game where there are 6 doors, there is a car behind 2 doors and there are goats behind 4 doors. I don't know what is behind the doors but I want a car. Let's say I pick a door at random, so Initially, my chances of winning a car are 1/3. But before I open my door to see if I won, the host of the game opens...

1) Two teams, A and B, are playing a best of 5 game series. (The series...

1) Two teams, A and B, are playing a best of 5 game series. (The series is over once one team wins 3 games). The probability of A winning any given game is 0.6. Draw the tree diagram for all possible outcomes of the series. 2) List all possible combinations of rolling a 4-sided die (d4) and a 6-sided die (d6) (enumaration). Also determine the probability X {1..6} where X is the largest of the two numbers. Two players, A...

. Two friends are playing a matching card game where friend 1, Max, chooses an Ace,...

. Two friends are playing a matching card game where friend 1, Max, chooses an Ace, Two, or Three and friend 2, Lucy, plays King, Queen, or Jack. They both put $ 5 each into the pot. Depending on what cards they play, they split the pot differently. The payoffs are summarized in the following table: Lucy King Queen Jack Max Ace (1,3) (3,5) (2,4) Two (6,5) (3,3) (3,2) Three (4,2) (5,4) (3,1) (a) Use iterated elimination of strictly...

Subjects