Problem 27 - 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 the jackpot by matching all five numbers drawn from white balls \((1 \text { through } 56\) ) and matching the number on the gold Mega Ball \((1 \text { through } 46) .\) What is the probability of winning the jackpot?

Step-by-Step Solution

Answer

The probability of winning the Mega Millions jackpot is \( \frac{1}{C(56, 5) * C(46, 1)} \)

Step 1: Calculate the combinations for white balls

First, we need to figure out how many ways we can choose 5 numbers out of 56 since in Mega Millions, a player picks five different numbers from 56. This is a standard formula for combinations: \( C(n, k) = \frac{n!}{k!(n-k)!} \), where \(n\) is the number of items to choose from, \(k\) is how many we need to choose and \( ! \) denotes factorial. So for this instance it would be: \( C(56, 5) = \frac{56!}{5!(56-5)!} \).

Step 2: Calculate the combinations for gold Mega Ball

The player also picks a number from 46 for the gold Mega ball. So, the combination would be \( C(46, 1) = \frac{46!}{1!(46-1)!} \). Considering the fact that order doesn't matter for jackpot winning and for each combination of white balls there is multiplied the choices for golden balls, we should multiply these two results.

Step 3: Calculate the probability of winning

The probability of winning is basically how many successful outcomes (which is 1, winning the jackpot) are, divided by total outcomes (which is the count calculated in step 1 and step 2). Using the notion P(A) is the probability of event A happening, we will get, \( P(win) = \frac{1}{C(56, 5) * C(46, 1)} \).

Key Concepts

Combinations in Probability

Understanding combinations in probability is essential when tackling problems such as calculating the lottery odds. In probability, a combination is a way of selecting items from a collection, such that the order of selection does not matter. In the context of lottery games like Mega Millions, this is crucial because it does not matter what order the numbers are drawn; all that matters is if you have the selected numbers.

To calculate the number of combinations, we use the formula:
\( C(n, k) = \frac{n!}{k!(n-k)!} \)
where \( n \) is the total number of items you can choose from and \( k \) is the number of items you choose. The exclamation point \( ! \) represents factorial notation, which we will elaborate on in the next section. In the lottery example, n would be the total numbers in the draw, and k would be the number of picks you are allowed.

For instance, picking 5 numbers from 56 without regard to the order can be represented as \( C(56, 5) \). This combination calculation determines the various ways you can win based on how many numbers you must match to win the lottery. Understanding the concept of combinations helps you grasp the complexity of lottery odds and is a fundamental principle in probability theory. It also illustrates why winning the lottery is such a rare event.

Factorial Notation

Factorial notation plays a paramount role in calculating combinations in probability theory. The notation represented by an exclamation mark \( ! \), signifies the product of a series of descending positive integers.

For example, \( 5! \) (read as 'five factorial') would be calculated as:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
The concept of factorial is essential when determining the number of possible arrangements in a set and, by extension, is used to compute combinations. Factorials can grow to be very large numbers quite quickly, which is why lottery odds are so immense—because you are calculating combinations of large sets of numbers.

The notation is applied within the formula for combinations. As seen in the lottery problem, to determine the number of possible combinations of 5 numbers from 56, we use the expression \( 56! \). The factorial expression expands to a very large number because it implies a multiplication series from 56 down to 1. This element of the combination equation contributes to the astronomically high number of possible outcomes in a lottery draw.

Probability as a Fraction

Probability, the measure of the likelihood of an event occurring, can be expressed as a fraction. In this context, the fraction is comprised of the number of favorable outcomes over the number of all possible outcomes.

When calculating the probability of winning the jackpot in the Mega Millions lottery, for instance, there is only one favorable outcome (the winning combination), while the total possible outcomes are the number of different combinations that can be drawn. Thus, the probability is:
\( P(win) = \frac{1}{C(56, 5) \times C(46, 1)} \)
By substituting the combination values into this fraction, you determine the precise odds of winning. Probabilities as fractions are particularly intuitive because they clearly show how one 'winning scenario' relates to the multitude of possible combinations. For many, visualizing probability as a fraction helps simplify an otherwise abstract concept and conveys the rare nature of certain events, such as winning the lottery. The smaller the fraction, the less likely the event is to occur, which is aptly demonstrated by the minuscule odds of securing a lottery jackpot.