Question

Free groceries. To attract shoppers, a supermarket runs a weekly contest that involves "scratch-off" cards. With each purchase, customers get a card with a black spot obscuring a message. When the spot is scratched away, most of the cards simply say, "Sorry-please try again." But during the week, 100 customers will get cards that make them eligible for a drawing for free groceries. Ten of the cards say they may be worth \(\$ 200,10\) others say \(\$ 100,20\) may be worth \(\$ 50\), and the rest could be worth \(\$ 20\). To register those cards, customers write their names on them and put them in a barrel at the front of the store. At the end of the week the store manager draws cards at random, awarding the lucky customers free groceries in the amount specified on their card. The drawings continue until the store has given away more than \(\$ 500\) of free groceries. Estimate the average number of winners each week.

Short answer

Approximately 12 cards are expected to be drawn each week to exceed the $500 giveaway limit.

Step-by-step solution

Understand the card distribution

Identify how many cards correspond to each prize. There are 100 eligible cards in total. They are distributed as follows: 10 cards for $200, 10 cards for $100, 20 cards for $50, and 60 cards for $20.

Calculate the total weekly giveaway limit

The total amount of groceries that will be given away each week is more than $500.

Define possible outcomes for each prize category

For each card prize value (i.e., $200, $100, $50, $20), calculate the total prize value if all corresponding cards were drawn. This helps understand the worst-case distribution scenario.

Calculate expected number of cards drawn

Given the cards and their values, calculate the expected number of cards drawn until more than $500 is given away. Use the available card values to compute probable combinations exceeding the $500 limit.

Determine expected value formula

For calculating the expected value, consider the probability of each card value being drawn multiplied by its respective card quantity, added to the cumulative value until the total exceeds $500.

Compute the expected number of winners

Utilize the expected number to approximate how many purchases lead to winners each week, given the drawing continues until more than $500 in prizes are distributed.

Key concepts

Random Variables

Random variables are fundamental concepts in probability and statistics. They represent outcomes of a random experiment, which might take on different values. In the context of our supermarket exercise, the random variable could be defined as the value of the prize drawn from the scratch-off card. This value changes unpredictably but follows a certain distribution of possibilities.

Each card has a designated probability of winning, found by dividing the number of each type of prize card by the total number of eligible cards. In this case:

• 10 cards have a prize of \( \\(200 \)
• 10 cards have \( \\)100 \)
• 20 cards have \( \\(50 \)
• 60 cards could be worth \( \\)20 \)

Understanding random variables allows us to analyze the likelihood of obtaining each prize, which is crucial for calculating the expected value and determining the number of winners.

Expected Value

The expected value, often seen as the 'average' outcome of a random experiment, gives us insight into the long-term behavior of a random variable. In this scenario, the expected value would guide us in choosing the average number of people who would win prizes until the total giveaway exceeds \( \$500 \).

The expected value can be found using the formula: \[ E(X) = \sum_{i=1}^{n} P(X_i) \cdot X_i \] where \( P(X_i) \) represents the probability of each prize value and \( X_i \) is the prize amount.

After we've calculated the expected value for each prize, we sum these values to get a comprehensive expected result across all possibilities. This is crucial, as it offers a clear picture of the most probable outcome over repeated contests.

Combinatorics

Combinatorics involves counting, arranging, and combination of elements within a set, making it a valuable tool for solving probability problems. In our grocery contest scenario, combinatorics helps determine the different ways in which the cards can be drawn without replacement until the prize threshold of \( \\(500 \) is exceeded.

Specifically, we are tasked with figuring out how different combinations of prize cards can be combined to achieve a total value beyond \( \\)500 \).

• Start by considering combinations starting with higher-value prizes, as they quickly contribute to reaching the target amount.
• Progress to including more of the lower-value cards to fill up the remaining amount once higher-value cards are used.

By understanding the permutations and combinations of the eligible prize cards, we can estimate an efficient and systematic sketch of potential outcomes, helping refine our expected value calculations.

Card Drawing

Card drawing in this context refers to the random selection process of choosing scratch-off cards that specify different amounts of prize money. It is essential to grasp how the card drawing contributes to the probability and expected value calculations.

The act of drawing a card can be thought of as a sequence of independent events, where each draw can result in winning one of the prize amounts based on the issued cards. The likelihood of drawing any specific prize reflects both the strategy for maximizing prize money distribution and the random nature of selection.

This aspect also considers continuity in the drawings until a certain prize collection limit (\( \$500 \)) is surpassed. Understanding this context ensures a thorough comprehension of how repeated drawings lead to a greater number of winners and assists in approximating an average number of prize claims per week. This comprehensive approach underpins the application of probability theory in practical, real-world scenarios.