Problem 12 - Algebra and Trigonometry

Question

Sweepstakes \(A\) sweepstakes offers a first prize of \(\$ 1,000,000,\) second prize of \(\$ 100,000,\) and third prize of \(\$ 10,000\) . Suppose that two million people enter the contest and three names are drawn randomly for the three prizes. (a) Find the expected winnings for a person participating in this contest. (b) Is it worth paying a dollar to enter this sweepstakes?

Step-by-Step Solution

Answer

Expected winnings are \(\$0.555\). Not worth entering at \$1 cost.

Step 1: Define the Random Variable

Define the random variable \( X \) as the winnings for a person participating in this contest. There are four possible outcomes: winning the first prize of \(\\(1,000,000\), the second prize of \(\\)100,000\), the third prize of \(\\(10,000\), or winning nothing (\(\\)0\)).

Step 2: Assign Probabilities to Each Outcome

Since there are two million participants and one prize of each kind, the probability of winning the first prize is \(\frac{1}{2,000,000}\), the second prize is also \(\frac{1}{2,000,000}\), and the same for the third prize. The probability of not winning anything is \(1 - \frac{3}{2,000,000}\).

Step 3: Calculate Expected Value

The expected winnings (\(E(X)\)) can be calculated using the formula for expected value: \[ E(X) = \sum (\text{Probability of Outcome} \times \text{Value of Outcome}) \] Substituting the values, we get: \[ E(X) = \left( \frac{1}{2,000,000} \times 1,000,000 \right) + \left( \frac{1}{2,000,000} \times 100,000 \right) + \left( \frac{1}{2,000,000} \times 10,000 \right) + \left( \left(1 - \frac{3}{2,000,000}\right) \times 0 \right) \] Calculating each term, \[ E(X) = 0.5 + 0.05 + 0.005 = 0.555 \] Therefore, the expected winnings are \(\$0.555\).

Step 4: Evaluate Worth of Entering Sweepstakes

Since the expected winnings (\\(0.555) are less than the cost of entering (\\)1), it is not financially beneficial to participate in this sweepstakes based purely on expectation.

Key Concepts

Expected Value

The concept of expected value is crucial in probability theory. It gives us a way to determine the average outcome we can anticipate if an experiment is repeated many times. In the context of the sweepstakes contest, we calculate the expected winnings for a participant to decide whether entering the contest is a good idea financially.

When we talk about expected value, it is essentially the sum of all possible outcomes, each multiplied by their probability of occurring. Mathematically, this is expressed as:

• \(E(X) = \sum (\text{Probability of Outcome} \times \text{Value of Outcome})\)

For our sweepstakes, we calculated that the expected value \(E(X)\) is \\(0.555. This means that on average, a participant can expect to win 55.5 cents, which is less than the \\)1 entry fee. Thus, the expected value helps us make informed decisions by showing that the likely value of participating is less than the cost of entering.

Random Variable

In probability theory, a random variable is a variable that takes on different values based on the outcome of a random event. It transforms possible outcomes into numerical values. This is fundamental for calculating probabilities, expected values, and variances.

For the sweepstakes example, the random variable \(X\) represents the amount of money won by a participant. Here, \(X\) can have one of several values, which are the prize amounts or zero in case the participant wins nothing.

• The possible values for \(X\) are \\(1,000,000, \\)100,000, \\(10,000, and \\)0.
• These values correspond to winning first, second, third, or no prize, respectively.

Thus, conceptualizing and defining a random variable allows mathematicians and statisticians to model real-world scenarios, like this sweepstakes, in a structured way.

Probability of Outcome

The probability of an outcome is a measure of how likely it is for that outcome to occur. It is a fundamental aspect of statistics and probability theory. In any random experiment, different outcomes have different probabilities.

For the sweepstakes, since there are two million participants and only one of each prize, probabilities are calculated as follows:

• The probability of winning the first prize is \(\frac{1}{2,000,000}\).
• The second and third prizes also have a probability of \(\frac{1}{2,000,000}\) each.
• The probability of winning nothing is \(1 - \frac{3}{2,000,000}\), which accounts for all scenarios where no prize is won.

Understanding these probabilities is essential to calculate the expected value. It informs you about the risks and potential benefits of participating in events like this sweepstakes. By noting the probabilities, you know just how slim the chances are of winning any prize.