Question

Deniz is playing a game of rock-paper-scissors with Phil. The loser of the game will pay the winner $ 100 $100dollar sign, 100. In the game of rock-paper-scissors, each player chooses either rock, paper, or scissors simultaneously. A winner is decided by the following rules: rock beats scissors, scissors beats paper, and paper beats rock. If both players make the same choice, then the game is a tie. Deniz is trying to decide whether to play rock or paper. Based on previous games, she knows that Phil will make his choice according to the probabilities in the table below. What are Deniz's expected winnings if she chooses rock? Round your answer to the nearest dollar. $ $dollar sign What are Deniz's expected winnings if she chooses paper? Round your answer to the nearest dollar. $ $dollar sign If Deniz wants the best payoff in the long run, she should choose . Phil’s choice Probability Rock 50 % 50%50, percent Paper 20 % 20%20, percent Scissors 30 % 30%30, percent

Answer 1

If Deniz chooses rock, her expected winnings are $20. If she chooses paper, her expected winnings are $30. To maximize her payoff in the long run, Deniz should choose paper.

Step-by-step explanation:

Deniz's expected winnings can be calculated by considering the probability of each outcome based on Phil's choices. If Deniz selects rock, she will win $100 if Phil chooses scissors (probability = 30%) and lose $100 if Phil selects paper (probability = 20%). If both players choose rock, it's a tie, so the winnings are $0 (probability = 50%).

The calculation for the expected winnings when Deniz chooses rock is:

`\[ \text{Expected winnings} = (100 * 0.30) + (-100 * 0.20) + (0 * 0.50) = 30 - 20 + 0 = 10. \]`

Rounding this to the nearest dollar gives an expected winning of $10 when Deniz selects rock.

Similarly, if Deniz chooses paper, she will win $100 if Phil chooses rock (probability = 50%) and lose $100 if Phil selects scissors (probability = 30%). A tie occurs if both choose paper (probability = 20%).

The calculation for the expected winnings when Deniz chooses paper is:

`\[ \text{Expected winnings} = (100 * 0.50) + (-100 * 0.30) + (0 * 0.20) = 50 - 30 + 0 = 20. \]`

Rounding this to the nearest dollar gives an expected winning of $20 when Deniz selects paper.

Therefore, based on the calculations, Deniz's expected winnings are higher when she chooses paper ($20) compared to choosing rock ($10). So, to maximize her payoff in the long run, Deniz should choose paper.

Answer 2

Deniz's expected winnings if she chooses rock are $30. If she chooses paper, her expected winnings are $20. If Deniz wants the best payoff in the long run, she should choose paper.

Expected winnings for rock:

Deniz wins against scissors with probability of 30%, and loses against paper with probability of 20%, so her expected winnings are (0.3100) - (0.2100) = 30

Expected winnings for paper:

Deniz wins against rock with probability of 50%, and loses against scissors with probability of 30%, so her expected winnings are (0.5100) - (0.3100) = 20