Problem 21 - Algebra and Trigonometry

Question

In the 6/49 lottery game, a player selects six numbers from 1 to 49. What is the probability of picking the six winning numbers?

Step-by-Step Solution

Answer

The probability is \( \frac{1}{13,983,816} \).

Step 1: Understand the Problem

In the 6/49 lottery game, a player needs to pick 6 numbers from a possible set of 49 numbers. We want to find the probability that a player picks the exact 6 winning numbers.

Step 2: Calculate Total Possible Combinations

The number of ways to choose 6 numbers from a set of 49 numbers is given by the binomial coefficient \( \binom{49}{6} \). It represents the total number of combinations possible in the game.

Step 3: Apply the Binomial Coefficient Formula

Calculate the binomial coefficient using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n = 49 \) and \( k = 6 \). Thus, \( \binom{49}{6} = \frac{49!}{6!(49-6)!} \).

Step 4: Simplify the Calculation

Substitute and simplify to find \( \binom{49}{6} = \frac{49 \times 48 \times 47 \times 46 \times 45 \times 44}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 13,983,816 \).

Step 5: Calculate the Probability

The probability of winning is the ratio of the favorable outcomes to the total possible outcomes. Since there is only 1 favorable outcome (the actual winning numbers), the probability is \( \frac{1}{13,983,816} \).

Key Concepts

binomial coefficient

The binomial coefficient is a fundamental concept in combinatorics. It describes the number of ways to choose a subset of items from a larger set, without considering the order of the items.
This is especially useful in problems where you are choosing a certain number of items from a pool, such as lottery games or team selection.

• Notation: The binomial coefficient is denoted as \( \binom{n}{k} \), pronounced "n choose k". Here, \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose.
• Formula: The formula for calculating the binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). The "!" symbol stands for factorial, which is the product of all positive integers up to that number. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

The binomial coefficient simplifies problems of determining how many ways you can choose \( k \) items from \( n \). It's pivotal for calculating combinations in various statistical and probability contexts.

lottery probability

Lottery probability refers to the likelihood of winning a lottery game by correctly guessing the winning numbers. In a typical lottery format like the 6/49, the calculations depend heavily on combinatorics.

• Total Possibilities: In a 6/49 lottery, there are \( \binom{49}{6} \) possible combinations of numbers that can be chosen. This number represents the huge total of possible selections one might make.


• Favorable Outcome: Out of all these possible combinations, only one is the winning combination—the one that matches the draw exactly.


• Probability Calculation: The probability of picking this exact set is \( \frac{1}{13,983,816} \), calculated by dividing the one winning outcome by the total number of possible outcomes.

Winning the lottery is exceedingly unlikely due to the vast number of combinations. Understanding this probability underscores the rare nature of hitting the jackpot.

combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and listing elements within a set. It's an essential part of probability theory and helps solve problems where there are multiple choices.

• Fundamental counting principle: This principle allows you to multiply the number of ways multiple, independent events can occur. For instance, if you have 3 shirts and 4 pants, you have \( 3 \times 4 = 12 \) outfit combinations.


• Combinations vs. Permutations: Combinatorics involves both combinations (where order doesn't matter) and permutations (where order does matter). Lotteries use combinations because the order in which the numbers are drawn doesn't matter.


• Applications: Beyond lotteries, combinatorics applies to things like graph theory, coding theory, and optimizing network connectivity.

Understanding combinatorics gives the tools to solve many puzzles and problems systematically, ensuring no possibilities are overlooked and probability calculations are accurately done.