Problem 1 - Algebra and Trigonometry

Question

If a game gives payoffs of \(\$ 10\) and \(\$ 100\) with probabilities 0.9 and \(0.1,\) respectively, then the expected value of this game is \(E=\) _____ \(\times 0.9+\) _____ \(\times 0.1\)

Step-by-Step Solution

Answer

The expected value is 19.

Step 1: Identify the Payoffs

In the exercise, there are two possible payoffs given: $10 and $100. These are the amounts you could win, depending on the outcome of the game.

Step 2: Identify the Probabilities

The probability corresponding to the $10 payoff is given as 0.9, and the probability for the $100 payoff is 0.1.

Step 3: Set Up the Expected Value Formula

The expected value of a game with different outcomes can be calculated using the formula: \[ E = (\text{payoff}_1 \times \text{probability}_1) + (\text{payoff}_2 \times \text{probability}_2) \] In this exercise, substitute the payoffs and probabilities into this formula.

Step 4: Substitute the Values

Substitute \( \\(10 \) for \( \text{payoff}_1 \), \( 0.9 \) for \( \text{probability}_1 \), \( \\)100 \) for \( \text{payoff}_2 \), and \( 0.1 \) for \( \text{probability}_2 \) into the formula. This yields:\[ E = (10 \times 0.9) + (100 \times 0.1) \]

Step 5: Calculate the Expected Value

Now perform the calculations:1. \( 10 \times 0.9 = 9 \)2. \( 100 \times 0.1 = 10 \)Add these two results together:\[ E = 9 + 10 = 19 \]

Key Concepts

Understanding Probability

Probability helps us determine the likelihood that a specific event will occur.
It's expressed as a number between 0 and 1.

• A probability of 0 means that the event is impossible.
• A probability of 1 means that the event is certain.
• Any value between 0 and 1 indicates the level of likelihood of the event occurring, with values closer to 1 signifying higher likelihood.

In our exercise, we have probabilities for different outcomes of a game:

• A 90% chance, or 0.9 probability, of receiving a $10 payoff.
• A 10% chance, or 0.1 probability, of winning a $100 payoff.

The concept of probability is essential in games and gambling, as it allows us to quantify and predict results.
This understanding can guide decision-making based on the likelihood of various outcomes.

Exploring Payoffs

Payoffs are the possible winnings or outcomes you may receive from a game or a decision.
They usually come in the form of monetary winnings, but can also represent other rewards. When calculating expected value, identifying all the potential payoffs is crucial.
In our scenario, we have two possible payoffs:

• $10, received with a probability of 0.9
• $100, received with a probability of 0.1

The notion of payoffs intertwined with their corresponding probabilities allows us to estimate the average payoff of a game over repeated plays.
By understanding payoffs, you can better analyze the potential rewards and risks involved in various scenarios.

Simplifying Mathematical Calculations for Expected Value

The expected value is a key concept to understand the potential average result of random events over time.
It is calculated as a weighted average of possible outcomes, where each outcome's weight is its probability.To calculate expected value:

• Identify each possible payoff and its probability.
• Multiply each payoff by its associated probability.
• Sum all these results.

For our example, the formula is:\[E = (10 \times 0.9) + (100 \times 0.1)\]Perform the calculations:

• For \(10 outcome: \(10 \times 0.9 = 9\)
• For \)100 outcome: \(100 \times 0.1 = 10\)

Adding these two results gives us the expected value:\[E = 9 + 10 = 19\]Simplifying these calculations helps us grasp the expected value's significance, illustrating the average winnings of a game played repeatedly. This concept is widely used in finance, insurance, and decision-making to conceptualize potential benefits or costs associated with uncertainty.