Problem 14 - 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

The expected winnings are \$0.555, hence it's not worth paying \$1 to enter.

Step 1: Identify the Number of Participants and Prizes

There are two million participants and three prizes in the sweepstakes: - First Prize: \(\\( 1,000,000\). - Second Prize: \(\\) 100,000\). - Third Prize: \(\$ 10,000\).

Step 2: Determine Probability of Winning Each Prize

The probability of winning any of the prizes is equal for each participant since the drawing is random. - Probability of first prize: \(\frac{1}{2,000,000}\). - Probability of second prize: \(\frac{1}{2,000,000}\). - Probability of third prize: \(\frac{1}{2,000,000}\).

Step 3: Calculate Expected Value of Winnings

The expected value (EV) is calculated by multiplying the probability of winning each prize by the prize amount and summing these products: \[EV = \left(\frac{1}{2,000,000} \times 1,000,000\right) + \left(\frac{1}{2,000,000} \times 100,000\right) + \left(\frac{1}{2,000,000} \times 10,000\right)\]This simplifies to: \[EV = 0.5 + 0.05 + 0.005 = 0.555\]

Step 4: Evaluate if it's Worth Paying to Enter

To determine if it's worth paying \(\\(1\) to enter, compare the expected value of \(\\)0.555\) to the cost of \(\\(1\). Since \(\\)0.555 \lt \\(1\), the expected loss per participant is \(\\)0.445\).

Step 5: Conclusion

Based on the expected value, participating in the sweepstakes is not worth paying \(\\(1\) since the expected winnings are \\)0.555, which is lower than the entry cost.

Key Concepts

Understanding Probability

Probability is at the heart of determining the outcome of a sweepstakes, or any event where chance plays a role. It measures the likelihood of a specific event happening. In sweepstakes, each participant gets an equal shot at winning, often expressed in terms of odds or probability. For example, if there are two million participants and only one person can win the first prize, the probability of any individual winning that prize is \[ P(\text{First Prize}) = \frac{1}{2,000,000} \]This means each person has one chance in two million to win the first prize. It's important to understand these low probabilities in sweepstakes, so expectations can be aligned with reality. This approach applies for the second prize and third prize too, where each has the same winning probability of \[ \frac{1}{2,000,000} \].It becomes essential, especially in decision-making processes, such as whether to participate in the sweepstakes, to understand these probabilities actively play a role in expected value calculations.

What is a Sweepstakes?

Sweepstakes are promotional contests that offer prizes to participants chosen by random drawing. They're often used by companies to promote products and engage consumers. Unlike lotteries, sweepstakes don’t require a purchase to enter, though sometimes an entry fee might be required for promotional reasons. In the sweepstakes mentioned:

• Participants enter the contest hoping to win one of three prizes.
• The draw is random, ensuring fairness as everyone who enters has an equal opportunity to win.

Prizes vary, as do the number of entries, which affects the odds of winning. Sweepstakes are a form of opportunity for many, with the excitement stemming from the chance to win substantial prizes, like the example's million-dollar first prize. However, given the odds, entering sweepstakes should be seen as a fun activity rather than a guaranteed path to riches. Understanding how these odds and processes work can help manage expectations realistically.

Prize Distribution and Expected Value

In sweepstakes, prize distribution refers to how prizes are allocated among winners. In our example, with a first prize of \(1,000,000, a second prize of \)100,000, and a third prize of \(10,000, the expected value helps determine if entering is financially wise. The expected value is a calculated prediction of gain or loss. It considers all possible outcomes (prizes), their probabilities, and sums up each outcome weighted by its probability. Essentially, it gives an average outcome you might expect if you could repeat the sweepstakes multiple times.The expected value (EV) formula used was:\[ EV = \left(\frac{1}{2,000,000} \times 1,000,000\right) + \left(\frac{1}{2,000,000} \times 100,000\right) + \left(\frac{1}{2,000,000} \times 10,000\right) \]Breaking it down, each prize is multiplied by its chance of winning and summed:

• First Prize Contribution: \)0.5
• Second Prize Contribution: \(0.05
• Third Prize Contribution: \)0.005

The total expected value is \(0.555. Comparing this to the entry fee of \)1, it becomes evident that the risk of not winning outweighs the small expected monetary gain, suggesting it might not be worth entering for financial gain alone.