Question

Consider the following game. Beginning with a deck of even size =N cards that are half red and half green ( N/2 each), you begin drawing cards. For each red card, you win $1 and for each green card you lose $1. You may stop at any time. What is the value of this game to you assuming you play optimally? Write a program that computes this value.

1>#!/ bin/python3 ...
10
11#
12 \# Complete the 'game_value' function below.
13#
14 \# The function is expected to return a DOUBLE.
15 \# The function accepts INTEGER deck_size as parameter.
16 \# deck_size is an even non-negative integer
17#
18
19 def game_value(deck_size):
20 \# Write your code here

Answer 1

The optimal strategy in the described card game, where red cards earn $1 and green cards result in a $1 loss, ensures a net value of zero, as the initial deck is evenly split. The provided Python program defines a function, `game_value`, that returns this expected value of zero for any even deck size.

The value of this game, assuming you play optimally, is always zero. This is because the deck is initially evenly split between red and green cards, and for each red card won, you lose $1 for each green card. Since the deck is symmetric, on average, you win and lose the same amount, resulting in a net value of zero.

Here's the completion of the Python program:

```python

def game_value(deck_size):

# The value of the game is always zero

return 0.0

# Example usage:

deck_size = 10

result = game_value(deck_size)

print(f"The value of the game with a deck size of {deck_size} is: {result}")

```

This simple program defines the `game_value` function that takes the deck size as input and always returns 0.0, indicating the expected value of the game is zero.

Answer 2

The value of the card-drawing game depends on the expected value and optimal stopping strategy. At the start, the expected value is $0 due to equal chances of winning or losing. A program to calculate the game value would consider changing probabilities and simulate the game for the optimal stopping point.

Step-by-step explanation:

The value of the card-drawing game can be understood through the concept of expected value, which is the long-term average outcome of a random process. In the scenario where the deck is of even size N, with equal numbers of red and green cards, and the player aims to play optimally, the optimal strategy would typically be to stop when ahead, as eventually, the outcomes will balance each other out. When N is even and the cards are equally split by color, the expected value at the start is $0 since there's an equal chance of drawing a red card (+$1) or a green card (-$1).

However, as cards are drawn and not replaced, probabilities change and the optimal stopping point can be calculated based on the deck's composition at any given moment. Writing a program to calculate the value of the game would involve using conditional probabilities and simulating the game to determine the optimal point to stop drawing cards. The actual code implementation would vary based on programming language and specific approach used.