Question

Find the probability of winning a lottery using the given rules. Assume that lottery numbers are selected at random. You must correctly select 4 numbers, each an integer from 0 to 9 . The order is important.

Short answer

The probability of winning the lottery, given the mentioned conditions, is \(1/10,000\) or \(0.0001\).

Step-by-step solution

Determine total possible outcomes

The first step is to determine how many outcomes are possible. In this lottery, four digits are selected. However, since each digit can be any number from 0 to 9, each digit has 10 possible outcomes. This is calculated as \(10 * 10 * 10 * 10\), or \(10^4\), which is equal to 10,000. Hence, there are a total of 10,000 possible outcomes.

Identify the number of successful outcomes

For a ticket to be a winner, the numbers must be chosen exactly. Hence, there is only one successful combination per ticket.

Apply the Probability formula

Probability refers to the likelihood that a particular event will occur, and it is calculated as the number of successful outcomes divided by the number of total possible outcomes. Given that there is only 1 successful combination and 10,000 total combinations, the probability formula will be applied as \(P(E) = \frac{{number of favorable outcomes}}{{total number of possible outcomes}} = \frac{1}{10,000}\).

Key concepts

Probability Formulas

The cornerstone of understanding lottery probabilities, or any random event's likelihood, lies in grasping the fundamentals of probability formulas. Probability measures the chance that an event will occur and it is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

The general probability formula is:
\[\begin{equation} P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} \end{equation}\]
In the context of the lottery problem, to find the probability of selecting the correct sequence of numbers, you consider the single favorable outcome – your chosen sequence – versus the total potential combinations. With a fundamental understanding of probability and its formula, you'll be better equipped to tackle various problems involving chance.

Permutations and Combinations

When it comes to lotteries and games of chance, 'permutations' and 'combinations' are part of the mathematical toolkit used to solve probability questions. A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. Any sequence of 4 lottery numbers is a permutation because the order is important.

Conversely, a combination is an arrangement where order does not matter. If our lottery were concerned with simply choosing 4 correct numbers regardless of order, we'd use combinations to calculate the chances.

Understanding the difference between these two concepts is critical when solving probability problems, as it will help you determine the correct approach for calculating the likelihood of an event.

Mathematical Probability Calculation

Mathematical probability calculation is a precise method of determining the likelihood of a particular event. In our lottery example, we calculate the total number of possible outcomes (permutations of four digits from 0 to 9, which is 10,000) and then determine the number of successful outcomes that will result in a win (which is 1).

By employing the probability formula, we can calculate the chance of winning the lottery with a single ticket as follows:
\[\begin{equation} P(\text{winning}) = \frac{1}{10,000} \end{equation}\]
This calculation inherently assumes that every outcome is equally likely, a key principle in problems of random selection. With a strong grasp of mathematical probability calculation, students are equipped to evaluate the risk and reward in a variety of scenarios beyond just lotteries.