Problem 23 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

LOTTERY The probability of winning \(\$ 100\) in a particular lottery is \(0.08\), the probability of winning \(\$ 20\) is \(0.12\), the probability of winning \(\$ 5\) is \(0.2\), and the probability of losing is \(0.6\). What is a fair price to pay for a lottery ticket?

Step-by-Step Solution

Answer

The fair price for a lottery ticket is $11.40.

Step 1: Define the Random Variable

Let the random variable X represent the prize won from the lottery ticket. The possible values of X are 100, 20, 5, and 0 (for losing).

Step 2: List the Probabilities

The given probabilities are: - P(X = 100) = 0.08 - P(X = 20) = 0.12 - P(X = 5) = 0.20 - P(X = 0) = 0.60

Step 3: Calculate Expected Value

To find a fair price, calculate the expected value (E) of X using the formula: \(E(X) = \text{sum of each outcome} \times \text{its probability}\). Thus, \(E(X) = (100 \times 0.08) + (20 \times 0.12) + (5 \times 0.20) + (0 \times 0.60)\)

Step 4: Compute Each Term

\(100 \times 0.08 = 8\) \(20 \times 0.12 = 2.4\) \(5 \times 0.20 = 1\) \(0 \times 0.60 = 0\)

Step 5: Sum the Expected Values

Add all the terms together to find the total expected value. \(E(X) = 8 + 2.4 + 1 + 0 = 11.4\)

Step 6: Determine Fair Price

The fair price for the lottery ticket is the expected value of winnings, which is \(11.40. Therefore, paying \)11.40 per ticket ensures fairness in the long run.

Key Concepts

random variable

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. In the context of our lottery example, the random variable is represented by the different amounts of money that could be won from a lottery ticket, including the possibility of winning nothing. These outcomes (100, 20, 5, and 0) are considered our random variable's values. It's crucial to define this variable clearly at the beginning, as it forms the basis for deciding the outcomes of our probability distribution.

probability distribution

Probability distribution describes how the probabilities are distributed over the values of the random variable. For our lottery scenario, we assign probabilities to each possible outcome:

• Winning \(100 has a probability of 0.08
• Winning \)20 has a probability of 0.12
• Winning \(5 has a probability of 0.20
• Losing (winning \)0) has a probability of 0.60

Each of these probabilities adds up to 1, ensuring that one of the outcomes must happen. By understanding the probability distribution, we can proceed to further calculate metrics such as the expected value, which provides more insight into what might happen on average.

fair price calculation

To determine a fair price for the lottery ticket, we need to compute the expected value of the ticket. The expected value represents the average amount of money we can expect to win per ticket in the long run. It integrates both the values of different outcomes and their respective probabilities. The step-by-step process is as follows:

First, identify the outcomes and their associated probabilities:

• \(100 with a probability of 0.08
• \)20 with a probability of 0.12
• \(5 with a probability of 0.20
• \)0 with a probability of 0.60

Next, multiply each outcome by its probability:

• (100 × 0.08 = 8)
• (20 × 0.12 = 2.4)
• (5 × 0.20 = 1)
• (0 × 0.60 = 0)

Finally, add all these products to get a total expected value, which represents a fair price to pay for the ticket.

expected value

The concept of expected value (E) plays a fundamental role in probability and statistics. It gives an average outcome if a random experiment is repeated many times. Expected value can be calculated using the formula:
E(X) = \text{sum of each outcome} \times \text{its probability}}In our lottery case, the expected value is calculated as:
E(X) = (100 × 0.08) + (20 × 0.12) + (5 × 0.20) + (0 × 0.60) = 8 + 2.4 + 1 + 0 = 11.4

The expected value tells us that if we played this lottery many times, on average, we would win \(11.4 per ticket. Hence, paying \)11.4 for a ticket is considered fair because this price represents the average winnings. Understanding the expected value ensures that both the seller and the buyer have a fair chance over many transactions.