Problem 51 - College Algebra with Modeling and Visualization

Question

To win a lottery, a person must pick 3 numbers from 0 to 9 in the correct order. If a number may be repeated, what is the probability of winning this game with one play?

Step-by-Step Solution

Answer

The probability of winning is \( \frac{1}{1000} \).

Step 1: Identify the total number of possible outcomes

Each number can be any digit from 0 to 9, and since there are 3 numbers that need to be picked, and each number can be picked independently of others, we calculate the total number of combinations as \(10 \times 10 \times 10 = 1000\). This represents all possible arrangements of numbers from 0 to 9 taken 3 at a time.

Step 2: Determine the number of favorable outcomes

A favorable outcome is one specific combination of numbers, as the player wants to win by picking the exact sequence. Since there's only one correct combination that wins the lottery, there is only 1 favorable outcome.

Step 3: Calculate the probability of winning

The probability of winning is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Hence, the probability is \( \frac{1}{1000} \).

Key Concepts

Lottery

Lotteries are games of chance where players strive to match a predetermined set of numbers in the correct sequence to win a prize. In the exercise mentioned, the player needs to pick a sequence of three numbers from 0 to 9 in the exact order they appear to win. Lottery games like these rely on randomness, and the huge number of possible combinations is what makes winning difficult.

• The appeal of lotteries hinges on the potential to receive large returns on a small investment.
• Understanding the probabilities involved helps manage expectations while playing.
• Lotteries are ubiquitous worldwide, often run by state or national organizations as a form of legal gambling that funds public services and projects.

While the odds might seem challenging, the thrill of potentially hitting the jackpot keeps many participants engaged.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and combining objects. In the context of the lottery exercise, combinatorics helps determine the number of potential lottery sequences.
We use the principle of independent counting, which states that each choice made is independent of others, to calculate the total number of possible combinations. Here, for each of the three numbers, there is a choice of 10 possibilities (0 to 9). Therefore, the total number of combinations is determined by multiplying the possibilities for each digit:

• First position: 10 choices
• Second position: 10 choices
• Third position: 10 choices
• Total combinations: \(10 \times 10 \times 10 = 1000\)

Mastering basic combinatorics principles allows one to solve problems that require understanding permutations, combinations, and arrangements effectively. This knowledge is crucial for fields like computer science, statistics, and operations research.

Number Sequences

Number sequences are ordered sets of numbers generated by specific rules or patterns. In the lottery, a sequence is the exact order of digits selected by the player. Understanding how these sequences work can provide insights into problems involving probabilities and arrangements.
In the exercise, the exact sequence of three numbers must be picked accurately to win, highlighting the importance of sequence order in combinatorial probability. Number sequences in lotteries differ from typical sequences, like arithmetic or geometric sequences, because:

• The order in which numbers appear is critical – switching the order creates a different sequence.
• Repetition is allowed within sequences, enabling more possible combinations.
• The exact match is necessary for a favorable outcome, as any deviation results in a loss.

Studying number sequences helps in understanding larger mathematical concepts and can be a valuable exercise in identifying patterns and predicting outcomes, especially in probability-related scenarios like lotteries.