Problem 9 - Algebra and Trigonometry

Question

Find the expected value (or expectation) of the games described.? $$ \begin{array}{l}{\text { A bag contains two silver dollars and eight slugs. You pay }} \\ {50 \text { cents to reach into the bag and take a coin, which you get }} \\ {\text { to keep. }}\end{array} $$

Step-by-Step Solution

Answer

The expected value is a loss of 30 cents per game.

Step 1: Identify the Outcomes

The possible outcomes when reaching into the bag are either picking out a silver dollar or a slug. Given that there are 10 coins total (2 silver dollars and 8 slugs), the number of outcomes is 2 for silver dollars and 8 for slugs.

Step 2: Determine the Probabilities

Calculate the probability of each outcome. The probability of picking a silver dollar is the number of silver dollars divided by the total number of coins. Thus, \( P(\text{Silver Dollar}) = \frac{2}{10} = 0.2 \). The probability of picking a slug is \( P(\text{Slug}) = \frac{8}{10} = 0.8 \).

Step 3: Assign Values to Outcomes

Assign a monetary value to each outcome. If you pick a silver dollar, you gain $1, but you paid 50 cents to play, leaving you with a net gain of 50 cents (or $0.50). If you pick a slug, it has no value, resulting in a net loss of 50 cents (or -$0.50).

Step 4: Calculate the Expected Value

Use the formula for expected value: \[ E(X) = \sum (P(x) \cdot x) \]where \( x \) represents the value of each outcome. Therefore, \[ E(X) = (0.2 \times 0.50) + (0.8 \times -0.50) \]Calculate each term: - First term: \( 0.2 \times 0.50 = 0.10 \)- Second term: \( 0.8 \times -0.50 = -0.40 \)Add these: \( 0.10 - 0.40 = -0.30 \).

Step 5: Conclusion: Interpret the Expected Value

The expected value of drawing a coin from the bag is \(-0.30\) dollars. This means that on average, you would lose 30 cents every time you play this game.

Key Concepts

Understanding Probability

Probability is a way of expressing the likelihood of an event occurring. In simple terms, it tells you how likely something is to happen. It's calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

In our game example, retrieving a silver dollar or a slug are the only two possibilities. To find the probability of picking a silver dollar from the bag, we look at how many silver dollars there are and how many total coins there are overall.

Given there are 2 silver dollars and 8 slugs, the probability of picking a silver dollar is:

• For a Silver Dollar: \( P(\text{Silver Dollar}) = \frac{2}{10} = 0.2 \)
• For a Slug: \( P(\text{Slug}) = \frac{8}{10} = 0.8 \)

These probabilities tell us that while there is a 20% chance of picking up a silver dollar, there is an 80% likelihood of picking a slug.

The Importance of Outcome Value

Each outcome in a game or experiment has a value, which is an essential factor when calculating expected values. Here, the outcome value is determined by the monetary gain or loss from each possible event.

Let's look at our two potential outcomes:

• Picking a Silver Dollar: You gain \(1, but since you paid \)0.50 to play, the net gain is \\(0.50.
• Picking a Slug: A slug is worthless. You still paid \)0.50 to play, so the net result is a loss of \$0.50.

Understanding these values is crucial because they are used as input when calculating the expected value. Outcome value affects decision-making as it provides insight into potential gains or losses from each outcome.

Expected Value Formula Explained

The expected value is a calculated average of all possible values. It provides a theoretical mean outcome if you were to repeat an experiment multiple times.

To find the expected value \( E(X) \), use the formula:
\[ E(X) = \sum (P(i) \cdot x(i)) \] where \( P(i) \) is the probability of each outcome and \( x(i) \) is the value of each outcome.

In the provided example, calculate the expected value as follows:

• First, consider the monetary gain for each outcome: \[ (0.2 \times 0.50) = 0.10 \]
• Then, factor in the loss from slugs: \[ (0.8 \times -0.50) = -0.40 \]

Finally, sum these results:
\[ E(X) = 0.10 - 0.40 = -0.30 \] This means, on average, you lose 30 cents per play. Through expected value calculations, you gain insight into the average result, helping guide decisions in uncertain situations.