Problem 6 - Algebra 2 and Trigonometry

Question

The buyer of a lottery ticket chooses four numbers from the numbers 1 to \(32 .\) Repetition is not allowed. a. How many combinations of four numbers are possible? b. What is the probability of choosing all four of the winning numbers?

Step-by-Step Solution

Answer

a. 35,960 combinations are possible. b. Probability is \( \frac{1}{35960} \).

Step 1: Understand the problem

We are choosing 4 different numbers from a set of 32 numbers, without repetition. We need to calculate how many combinations of these 4 numbers are possible and the probability of choosing the winning set of numbers.

Step 2: Calculate the number of combinations

Since repetition is not allowed, and the order does not matter, we use the combination formula \[ C(n, r) = \frac{n!}{r!(n-r)!} \]where \( n = 32 \) and \( r = 4 \). Plugging in the values, we have:\[ C(32, 4) = \frac{32!}{4!(32-4)!} = \frac{32 \times 31 \times 30 \times 29}{4 \times 3 \times 2 \times 1} = 35960 \]

Step 3: Calculate probability of winning

To win, you must pick exactly the 4 winning numbers. There is only one set of these numbers. The probability \( P \) is calculated as:\[ P = \frac{\text{Number of winning combinations}}{\text{Total number of combinations}} = \frac{1}{35960} \]

Step 4: Conclusion

There are 35,960 possible combinations of choosing 4 numbers out of 32. The probability of choosing all four winning numbers is very small, specifically \( \frac{1}{35960} \).

Key Concepts

Understanding Probability

Probability is a fundamental concept in mathematics that measures how likely an event is to occur. It's basically a way to quantify uncertainty, measuring how often different events happen compared to all possible events.
A probability ranges between 0 and 1, where 0 means an event will not happen, and 1 means it will definitely happen. In the context of our lottery problem, the event is choosing the winning set of numbers.

• A probability of 0 indicates an impossible event.
• A probability of 1 indicates a certain event.
• The probability of all possible outcomes is equal to 1.

Understanding probability is crucial because it allows us to predict the likelihood of events, such as winning the lottery, which can be quite exciting albeit often slim.

Introduction to the Combination Formula

The combination formula is used when you are choosing a set number of items from a larger pool, and the order of selection does not matter. It calculates the number of potential groups (or combinations) that can be made.
The formula is given by:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]

• Here, \( n \) is the total number of items to choose from.
• \( r \) is the number of items to choose.
• The exclamation mark (!) denotes a factorial, which is the product of an integer and all the integers below it.

In our lottery problem, the combination formula helps determine how many different sets of 4 numbers can be chosen from the numbers 1 to 32, making it an essential tool for solving problems involving selection without regard to order.

Calculating Lottery Probability

Calculating lottery probability means determining the likelihood of picking the correct combination from a set of possible combinations. This involves two key steps: finding the total number of combinations first and then determining the probability of winning.
Using the lottery scenario from our exercise:

• We calculated that there are 35,960 possible combinations of choosing 4 numbers from 32.
• Since there is only one winning combination, the probability is \( \frac{1}{35960} \).

This incredibly low probability illustrates just how difficult it typically is to win such games. However, understanding these calculations helps manage expectations and provides insight into the random nature of lotteries.