Problem 28 - Algebra and Trigonometry

Question

Mega Millions is a multi-state lottery played in most U.S. states. As of this writing, the top cash prize was \(\$ 656\) million, going to three lucky winners in three states. Players pick five different numbers from 1 to 56 and one number from 1 to \(46 .\) Use this information to solve Exercises \(27-30 .\) Express all probabilities as fractions. A player wins a minimum award of \(\$ 10,000\) by correctly matching four numbers drawn from white balls ( 1 through 56 ) and matching the number on the gold Mega Bali (1 through 46 ). What is the probability of winning this consolation prize?

Step-by-Step Solution

Answer

The probability of winning the minimum award of \$10000 in Mega Millions lottery is given by the formula \([C(5, 4) * C(51, 1) * C(46, 1)] / [C(56, 5) * C(46, 1)]\).

Step 1: Understanding the problem

In the Mega Millions lottery game, the player chooses 5 different numbers from 1 to 56 and one number from 1 to 46. This means that the total number of possible outcomes is the all possible combinations of choosing 5 numbers from 56, and one from 46. The player wins if they correctly match 4 numbers drawn from the first set and the number from the second set. We want to calculate the probability of this happening.

Step 2: Calculate total outcomes

To calculate total outcomes, multiply the number of combinations of choosing 5 numbers from 56, and one from 46. We use the 'combinations' equation \(C(n, k) = n! / (k!(n-k)!) \). For choosing 5 numbers from 56, we have \(C(56, 5)\). For choosing 1 number from 46, it is \(C(46, 1)\). So the total number of outcomes is \(C(56, 5) * C(46, 1)\).

Step 3: Calculate winning outcomes

The player wins if they match 4 numbers from the first set and the number from the second set. For the first part, they need to get 4 numbers correct and 1 number wrong from the first 5 numbers selected, which is given by \(C(5, 4) * C(51, 1)\). For the second part of the game, we need to correctly guess the Gold Mega Ball, which is given by \(C(46, 1)\), as there are 46 options and we need to choose 1 of them. So the winning outcomes are given by \(C(5, 4) * C(51, 1) * C(46, 1)\).

Step 4: Calculate the probability of winning

Probability is calculated as the number of desired outcomes over the total number of outcomes. So using the results from step 2 and step 3, we get the probability \(P\) as \(P = [C(5, 4) * C(51, 1) * C(46, 1)] / [C(56, 5) * C(46, 1)]\). This gives us the probability of winning a minimum award in Mega Millions lottery.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. It provides tools to calculate possible arrangements in a given scenario, such as the lottery. This is where the concept of combinations comes into play. When it comes to computing combinations, the key formula used is the combination formula, denoted as \(C(n, k) = \frac{n!}{k!(n-k)!} \). Here:

• \(n\) represents the total number of items available.
• \(k\) represents the number of items to be chosen.
• \(!\) signifies a factorial, which is the product of all positive integers up to a given number.

We use this formula to determine how many ways we can choose a specific number of items from a larger pool without regard to order. In the Mega Millions lottery, understanding combinatorics allows us to determine the total possible combinations of chosen lottery numbers.

Probability Calculation

Probability is a way to measure how likely an event is to happen. In the context of lotteries, we calculate probabilities to know what the chances are of certain outcomes, such as winning a prize. It is computed by dividing the number of successful outcomes by the total number of possible outcomes. The formula for probability is given by:

• \(P = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}\)

In the Mega Millions example, the probability calculation involves finding both the number of ways to achieve a successful outcome (i.e., winning the prize by picking 4 correct numbers from one set and the correct Gold Mega Ball from another set) and the total possible number of lottery combinations that can be chosen by the player. This requires understanding and utilizing the principles of both probability and combinatorics.

Mega Millions Lottery

The Mega Millions lottery is one of the most well-known multi-state lotteries in the United States. It's a game of chance that provides players with the opportunity to win substantial prizes. In this particular lottery structure, participants select five numbers from a pool of 56 white balls, along with one additional number, known as the Gold Mega Ball, from a separate pool of 46 numbers.
The rules are simple but achieving a win is challenging due to the multitude of number combinations that can be drawn. The lottery is based on a random selection process, and players can win prizes by matching different sets of numbers, with the jackpot being awarded for matching all the selected numbers and the Gold Mega Ball.
Understanding how the Mega Millions lottery works, including what numbers to choose and the odds involved, is crucial for players trying their luck at winning. Each combination's probability can be computed using mathematical tools like combinatorics to express these odds clearly.

Mathematical Modeling

Mathematical modeling involves creating mathematical formulas and equations to represent real-world scenarios. In the case of the Mega Millions lottery, a mathematical model can help to determine the probabilities of various outcomes. The model translates the rules of the lottery into mathematical terms using concepts like combinatorics and probability.
This often consists of identifying the relevant variables, such as the total numbers on the ball, the numbers drawn, and the probability of winning conditions. By applying these variables and combinatorial principles, a model is built to provide insights into lottery probabilities.
In the Mega Millions example, the mathematical model would consist of calculations involving combinations to count all possible outcomes, alongside probabilities to determine the likelihood of winning prizes from matching various numbers. Such models not only provide valuable insights but also help in making informed decisions when playing the lottery.