Problem 13 - Algebra and Trigonometry

Question

A Game of Chance \(A\) box contains 100 envelopes. Ten envelopes contain \(\$ 10\) each, ten contain \(\$ 5\) each, two are "unlucky," and the rest are empty. A player draws an envelope from the box and keeps whatever is in it. If a person draws an unlucky envelope, however, he must pay \(\$ 100\) . What is the expectation of a person playing this game?

Step-by-Step Solution

Answer

The expectation of playing this game is $-0.5$.

Step 1: Determine the Probabilities

First, determine the probabilities of drawing each type of envelope from the box.- Probability of drawing a \( \\) 10 \( envelope: \( \frac{10}{100} = 0.10 \) - Probability of drawing a \)\\( 5 \) envelope: \( \frac{10}{100} = 0.10 \)- Probability of drawing an unlucky envelope: \( \frac{2}{100} = 0.02 \)- Probability of drawing an empty envelope: \(1 - (0.10 + 0.10 + 0.02) = 0.78 \).

Step 2: Calculate Expected Value from Each Outcome

Next, calculate the expected value contribution from each type of envelope. This is done by multiplying the value of the envelope by its probability.- Contribution from a \( \\) 10 \( envelope: \(10 \times 0.10 = 1 \)- Contribution from a \) \\( 5 \) envelope: \(5 \times 0.10 = 0.5 \)- Contribution from an unlucky envelope: \(-100 \times 0.02 = -2 \)- Contribution from an empty envelope: \(0 \times 0.78 = 0 \).

Step 3: Sum the Expected Values

Add up all the contributions from each type of envelope to find the total expected value of the game:- Total Expected Value: \(1 + 0.5 - 2 + 0 = -0.5 \).

Key Concepts

Understanding Probability

Probability is the backbone of predicting outcomes in uncertain situations. It helps us understand how likely it is for a specific outcome to happen. In our "Game of Chance" example, probability is used to determine how often we might draw each type of envelope.
Here's a breakdown of how probabilities were calculated in the exercise:

• With 10 envelopes containing \(10 each, the probability of drawing one is \(0.10\) or 10%.
• Similarly, for the \)5 envelopes, the chance is also \(0.10\).
• For the unlucky envelope, it becomes \(0.02\) or 2%, given there are only 2 such envelopes.
• The rest are empty, so their probability naturally covers all remaining options at \(0.78\).

Each probability represents a fraction of the total possibilities. Combining these gives a full picture of potential outcomes. Recognizing these probabilities is crucial for calculating the expected value in decision-making.

Role of Game Theory

Game theory explores strategic interactions where outcomes depend on the actions of players. This game of chance is a basic example, illustrating a single player's decision-making process when facing random outcomes.
In such scenarios, the player assesses:

• The potential gains, like drawing a $10 or $5 envelope.
• The severe loss associated with drawing an unlucky envelope, resulting in a $100 penalty.
• The neutral outcome of drawing an empty envelope.

Game theory prompts us to think strategically about choices. By evaluating costs, benefits, and probabilities, the player weighs up options to decide whether the game is worth playing.
Understanding this framework can be vital in fields like economics and negotiations, where similar decision-making under uncertainty is required.

Introduction to Random Variables

In probability theory, a random variable represents numerical outcomes of a random phenomenon. Here, each envelope type in our game is associated with a random variable:

• $10 or $5 as successful outcomes.
• - $100 as a negative outcome (unlucky draw).
• $0 for an empty envelope.

These outcomes are not just numbers; they represent the gains or losses tied to drawing different envelopes. When calculating the expected value, each envelope type is multiplied by its probability, thus integrating the likelihood of this random variable.
In simple terms, random variables help us to quantify situations involving randomness. They provide a clear understanding of how different outcomes contribute to the overall result. Successfully working with random variables is key to making accurate predictions in probability and statistics.