Question

In Exercises 49 and 50 , find the probability of winning a lottery using the given rules. Assume that lottery numbers are selected at random. You must correctly select 6 numbers, each an integer from 0 to 49 . The order is not important.

Short answer

The probability of winning the lottery is \( \frac{1}{\binom{50}{6}} \). To calculate \( \binom{50}{6} = \frac{50!}{6!(50-6)!} \), then take the reciprocal. This will provide the probability of winning.

Step-by-step solution

Define the total number of possible outcomes

In a lottery where you have to select 6 numbers from a pool of 50 (0 to 49 inclusive), the total number of possible outcomes can be found by calculating the combination of 50 items taken 6 at a time. This can be represented mathematically by the combination notation \( C(n, r) \) or \( \binom{n}{r} \) which can be computed using the formula \( \frac{n!}{r!(n-r)!} \) where \( n \) is the total number of items, \( r \) is the number of items to choose, and \( ! \) represents the factorial of a number. Applying this formula, we find that the total number of outcomes \( = \binom{50}{6} = \frac{50!}{6!(50-6)!} \)

Calculate the probability of winning

Assumedly a 'win' means selecting the correct 6 numbers. The amount of ways to 'win' is 1; that is selecting the exact 6 numbers. Probability is thus calculated as favorable outcomes divided by total outcomes. In this scenario, the favorable outcome is one (since we're aiming to select the correct 6 numbers exactly). Therefore, the probability of winning the lottery is given by \( \frac{1}{\binom{50}{6}} \)

Simplify the calculation

To calculate \( \frac{1}{\binom{50}{6}} \), you can calculate \( \binom{50}{6} = \frac{50!}{6!(50-6)!} \), then take the reciprocal. The ! notation means factorial which is the product of all positive integers up to that number. For example, 5! = 5 * 4 * 3 * 2 * 1. The calculated probability will represent the odds of winning the lottery under these circumstances

Key concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is particularly used in situations where we want to know how many different ways there are to arrange a set of items, or how many different combinations of items are possible.

In our lottery scenario, we are dealing with the concept of combinations—a core idea in combinatorics—where the order in which the numbers are drawn does not matter. This is different from permutations, where the order is significant. To visualize combinatorics at play in the lottery, imagine each number as a unique marble, and each combination as a distinct way of picking 6 marbles out of a total of 50, without regard to the sequence in which they were chosen.

The realm of combinatorics is vast, but for our lottery example, understanding combinations provides a fundamental grasp of the core concept at hand. It demonstrates how we can systematically calculate the odds of winning in a scenario where selection order is irrelevant, but the selection itself holds the key to victory.

Factorial Notation

Factorial notation is a mathematical concept denoted by the exclamation mark (!) that represents the product of an integer and all the non-negative integers below it. To put it simply, if we have a number 'n', then its factorial, expressed as 'n!', would be the multiplication of all positive integers from 1 to 'n'.

For example, the factorial of 5, written as '5!', equals 5 * 4 * 3 * 2 * 1, which is 120. It's essential to note that the factorial of 0 is defined as 1, often surprising to those new to the concept. Factorial calculation comes into play in our lottery question within the formula used to determine the number of possible outcomes, highlighting its profound impact on combinatorial problems.

This factorial operation is crucial because it allows us to determine the size of large sets of combinations or permutations with relative ease, providing a foundation for a wide range of probability calculations, including those used in lottery drawings.

Probability Calculation

Probability calculation is a fundamental aspect of statistics, providing us with a way to quantify the likelihood of a particular event occurring. It takes into account the favorable outcomes over the total possible outcomes, often expressed as a fraction or percentage. In mathematical terms, this is shown as P(Event) = Number of Favorable Outcomes / Total Number of Possible Outcomes.

When calculating the probability of winning a lottery, as in the example provided, we use the combination formula to determine the total number of possible outcomes—this set the stage for finding our 'denominator' in the probability fraction. The 'numerator' is typically the number of favorable outcomes, which, for most lotteries where only one combination wins the jackpot, is 1.

Understanding the principles behind probability calculations helps determine not just our chances in games of chance, like the lottery, but also in predicting outcomes in more complex situations such as risk assessment, genetics, market trends, and beyond. It teaches us that while we can never be certain about the future, we can make informed guesses based on the structure of chance and patterns that emerge from our observations.