Problem 52 - College Algebra

Question

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting 20 numbers from the numbers 1 to 80 . After the player makes his selections, 20 winning numbers are randomly selected from numbers 1 to 80 . A win occurs if the player has correctly selected 3,4 , or 5 of the 20 winning numbers. (Round all answers to the nearest hundredth of a percent.) What is the percent chance that a player selects exactly 4 winning numbers?

Step-by-Step Solution

Answer

The percent chance is approximately 43.41%.

Step 1: Identify the Total Combinations

The total number of ways to select 20 numbers out of 80 is given by the combination formula \( C(n, k) \), where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose. Thus, the number of ways to choose 20 numbers from 80 is \( C(80, 20) \).

Step 2: Calculate the Combinations for Exact Matches

To find the number of combinations where a player selects exactly 4 winning numbers, we calculate: \( C(20, 4) \), which is the number of ways to choose 4 winning numbers from the 20 selected numbers, and \( C(60, 16) \), which is the number of ways to choose the remaining 16 winning numbers from the other 60 numbers (those not selected by the player).

Step 3: Calculate the Probability

The probability of the player selecting exactly 4 winning numbers is given by the formula: \( \frac{C(20, 4) \times C(60, 16)}{C(80, 20)} \). Calculate each component and simplify the fraction.

Step 4: Convert to Percentage

After calculating the probability, multiply the result by 100 to convert it from a probability to a percentage. Round the final answer to the nearest hundredth of a percent.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics concerned with counting, arranging, and finding patterns. It focuses on how items can be arranged or combined. In probability, combinatorics helps determine how likely certain outcomes are, by assessing the various combinations that can occur.

• **Counting Combinations:** Combinatorics is used to calculate the number of ways a certain selection can happen, such as how many ways you can choose a subset from a larger set.
• **Application in Games:** In games like Keno, which involve selecting numbers, combinatorics helps compute all possible ways numbers can be picked and determine winning combinations.

By understanding these combinations, players can estimate how likely they are to win, helping them make more informed decisions.

Keno

Keno is a popular lottery-style gambling game often found in casinos and online platforms. Players select numbers from a predefined range, typically between 1 and 80.

• **Selecting Numbers:** In Keno, players choose a set number, such as selecting 20 out of 80 possible numbers.
• **Winning:** Once selections are made, the game selects winning numbers randomly. Players win based on how many of their chosen numbers match the drawn numbers.

Keno combines elements of strategy and luck, as players decide how many numbers to choose and must rely on chance for winning numbers to align with their selections. Understanding the probability and combinatorics behind Keno can enhance your playing strategy.

Percent Chance

The percent chance describes the likelihood that a specific event will occur, expressed as a percentage. It's a crucial concept in probability theory that helps make predictions about outcomes.

• **Convert Probability to Percent:** Once you find the probability of an event, multiply that number by 100 to convert it into a percentage. This makes it easier to understand and compare with other probabilities.
• **Round Off:** It's common to round to the nearest hundredth of a percent for clarity in presentation, especially important in precise contexts like gambling.

Expressing probability as a percentage can give a clearer understanding of the odds, such as determining how likely you are to win a round of Keno.

Combination Formula

The combination formula is a fundamental tool in combinatorics used to determine how many ways you can choose items from a larger set without regard to order. The formula is represented as:\[ C(n, k) = \frac{n!}{k!(n-k)!} \]where:

• **\( n \):** The total number of items.
• **\( k \):** The number of items to be chosen.
• **\(!\):** Factorial, meaning the product of all positive integers up to that number.

In the Keno exercise, we use this formula to calculate different combinations:

• **Total Ways to Choose Numbers:** Calculate \( C(80, 20) \) to find all ways to select 20 numbers out of 80.
• **Exact Match Combinations:** Use \( C(20, 4) \) and \( C(60, 16) \) to see how many ways to select winning numbers from total options.

By applying the combination formula, you can easily assess the number of possible outcomes and work towards finding the probability of specific events, like hitting exactly 4 numbers on a Keno ticket.