Problem 28 - College Algebra

Question

A state lottery is designed so that a player chooses six numbers from 1 to 30 on one lottery ticket. What is the probability that a player with one lottery ticket will win? What is the probability of winning if 100 different lottery tickets are purchased?

Step-by-Step Solution

Answer

Probability of winning with one ticket is \(1 / C(30, 6)\); with 100 tickets, the probability is \(100 / C(30,6)\).

Step 1: Calculate Total Combinations

Using the formula of combination, determine the total number of possibilities where 6 numbers can be selected out of 30. \(C(30, 6) = 30! / 6!(30-6)!\)

Step 2: Determine the Probability of Winning with One Lottery Ticket

The player has 1 winning combination and, as calculated above, there are C(30, 6) total combinations. Thus, the probability of winning with one lottery ticket is \(1 / C(30, 6)\).

Step 3: Calculate the Probability of Winning with 100 Lottery Tickets

If 100 different lottery tickets are purchased, then there are 100 winning combinations. Thus, the probability of winning with 100 lottery tickets is \(100 / C(30, 6)\).

Key Concepts

Combinations and Permutations

Understanding the difference between combinations and permutations is fundamental when dealing with probability in games like lotteries. In simple terms, a combination is a selection of items where the order does not matter, whereas a permutation is a selection of items where the order does matter. Lottery games typically depend upon combinations because the order in which the numbers are drawn isn't important, only the set of numbers themselves.

For example, in our lottery exercise, a player chooses six numbers out of thirty. Using combinatorial terms, we are looking for the number of combinations, not permutations, because 1-2-3-4-5-6 is the same as 6-5-4-3-2-1 in a typical lottery draw.

Probability of Winning a Lottery

The probability of an event is a measure of how likely it is to occur, represented as a fraction between 0 and 1. In lottery games, the probability of winning is calculated by dividing the number of winning outcomes by the total number of possible outcomes.

In our exercise, the probability of winning with one ticket is incredibly small because there is only one winning combination against a vast number of possible combinations. As illustrated in the solution step 2, the formula \(1 / C(30, 6)\) yields the probability for one ticket. If a person buys 100 different tickets, the chance increases, but it is still relatively small, reflected by the calculation \(100 / C(30, 6)\) in step 3. The chances of winning do not increase linearly with the number of tickets purchased, which is a common misconception.

Factorial Notation in Probability

Factorial notation, denoted by an exclamation point (!), is essential in calculations involving probability, especially with lotteries. In mathematics, the factorial of a non-negative integer \(n\), denoted as \(n!\), is the product of all positive integers less than or equal to \(n\). It's a way to count how many different arrangements (permutations) there are.

For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). In the context of our lottery exercise, factorial notation is used within the combination formula. The total number of combinations, \(C(30, 6)\), relies on factorials to determine all possible groups of six numbers that can be drawn from thirty, which is calculated as \(30! / (6! \times (30-6)!)\).