Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Five hundred raffle tickets were sold. What is the probability that a person holding one ticket will win the first prize? What is the probability that he or she will not win the first prize?

Step-by-Step Solution

Answer

The probability of a person holding one ticket winning the first prize is \( P(win) = \frac{1}{500} \). The probability of a person holding one ticket not winning the first prize is \( P(not \thinspace win) = \frac{499}{500} \).

Step 1: Identify the total number of outcomes

In this raffle, 500 tickets were sold. This means that there are 500 possible outcomes, as each ticket has an equal chance of winning.

Step 2: Calculate the probability of winning the first prize

To find the probability of winning the first prize, we need to find the ratio of favorable outcomes to the total number of outcomes. In this case, the favorable outcome is holding the winning ticket, which is 1 (as there is only one first prize). So, the probability of winning the first prize is: \( P(win) = \frac{favorable \thinspace outcomes}{total \thinspace outcomes} = \frac{1}{500} \)

Step 3: Calculate the probability of not winning the first prize

To find the probability of not winning the first prize, we need to find the number of unfavorable outcomes (not winning) and divide it by the total number of outcomes. The number of unfavorable outcomes is the total number of tickets minus the winning ticket, which is \(500 - 1 = 499\). So, the probability of not winning the first prize is: \( P(not \thinspace win) = \frac{unfavorable \thinspace outcomes}{total \thinspace outcomes} = \frac{499}{500} \)

Step 4: Write the final answer

The probability of a person holding one ticket winning the first prize is: \( P(win) = \frac{1}{500} \) The probability of a person holding one ticket not winning the first prize is: \( P(not \thinspace win) = \frac{499}{500} \)

Key Concepts

Raffle Tickets

Raffle tickets are a popular way to randomly select winners for prizes in a competition. Each ticket serves as a unique entry for the holder in the raffle draw. When you buy a raffle ticket, you're essentially buying a chance to win a prize. Usually, the more tickets you have, the better your chances of winning, but each ticket by itself still represents a single chance.

In our scenario, 500 raffle tickets were sold. These tickets create 500 possible outcomes, as any one of them could be the winner. It's important to understand that the fairness of the raffle relies on each ticket having an equal chance of winning. This equal probability ensures that everyone who holds a ticket can potentially win the prize.

Favorable Outcomes

In probability, when we talk about favorable outcomes, we refer to the outcomes that align with the event we are interested in. For our raffle example, the favorable outcome is winning the first prize. This means holding the one winning ticket out of the 500 sold.

To calculate the probability, we take the number of favorable outcomes and compare it to the total number of possible outcomes. The formula for calculating the probability of a favorable event happening is:

• Number of favorable outcomes
• Divided by the total number of outcomes

In this raffle, the probability of winning is:
\( P(win) = \frac{1}{500} \)

This equation tells us that with one ticket, a person's chance of claiming the first prize is one out of five hundred.

Unfavorable Outcomes

Unfavorable outcomes are those that do not match the desired event. In our raffle, the unfavorable outcome is not winning the first prize. Since there is only one winning ticket, the rest fall into the unfavorable category.

To compute the probability of not winning, we subtract the one winning ticket from the total tickets, leaving 499 tickets that could result in losing. The unfavorable outcomes, therefore, amount to 499 in this scenario.

The formula for calculating the probability of not achieving the desired outcome is similar to that for favorable outcomes:

• Number of unfavorable outcomes
• Divided by the total number of outcomes

For our raffle, this is expressed mathematically as:
\( P(not \thinspace win) = \frac{499}{500} \)

This probability shows that if a person is holding one ticket, their chance of not winning is 499 out of 500, which is quite high.