Problem 64 - Algebra and Trigonometry

Question

Lottery In the 6\(/ 49\) lottery game, a player selects six numbers from 1 to 49 and wins if he or she selects the winning six numbers. What is the probability of winning the lottery two times in a row?

Step-by-Step Solution

Answer

The probability of winning the lottery two times in a row is \( \frac{1}{195,547,109,921,856} \).

Step 1: Calculate Probability of Winning Once

To win the 6/49 lottery, you need to choose the correct 6 numbers out of 49. This is a combination problem. The total number of possible combinations of 6 numbers from 49 is given by \( \binom{49}{6} = \frac{49!}{6!(49-6)!} \). After calculating, this equals 13,983,816 possible combinations. Therefore, the probability of winning the lottery once is \( \frac{1}{13,983,816} \).

Step 2: Calculate Probability of Winning Twice

To win the lottery twice in a row, you need to first win the lottery once, and then win again. The probability of these two independent events both occurring is the product of their probabilities. Therefore, the probability of winning twice in a row is \( \left( \frac{1}{13,983,816} \right)^2 = \frac{1}{195,547,109,921,856} \).

Key Concepts

Combination

In mathematics, a combination is a selection of items where the order does not matter. It's crucial in probability when determining potential outcomes for a given scenario.
A common example is in lotteries where you have to choose several numbers from a larger set.

• To calculate combinations, we use the formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).
• \( n \) is the total number of items to choose from, and \( r \) is the number of items to select.
• The exclamation mark (\( ! \)) denotes a factorial, which is the product of all positive integers up to that number.

Using combinations allows us to find out how many different groups of numbers can be selected, which is essential for calculations in contexts like lotteries.

Lottery Probability

Lottery probability refers to the odds of selecting the winning combination of numbers in a game. For example, in a typical 6/49 lottery, you pick 6 numbers from a set of 49.
This is a combination problem because the order of numbers doesn't matter.

• The total possible combinations are calculated using the combination formula: \( \binom{49}{6} \).
• In our example, this equals 13,983,816 combinations.
• The probability of selecting the winning combination is therefore \( \frac{1}{13,983,816} \).

Lottery games are designed to have very low probabilities of winning, making winning them a rare event.

Independent Events

In probability, independent events are those whose outcomes do not affect each other. Understanding this concept helps in scenarios where events happen repeatedly but independently.

• If two events A and B are independent, the probability of both occurring is the product of their individual probabilities.
• This can be written as \( P(A \text{ and } B) = P(A) \times P(B) \).
• Such calculations are applicable in cases like winning the lottery multiple times, where each attempt is independent of the previous ones.

Thus, for winning the lottery twice in a row, we multiply the probability of winning once by itself, illustrating the concept of independent events.

Probability Calculation

Probability calculation is the process of determining the likelihood of a particular outcome. It is expressed as a ratio of favorable outcomes to total possible outcomes.
Let's break down the basics:

• The formula for probability is \( P(Event) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \).
• In situations like our lottery example, favorable outcome is choosing the correct numbers, while total outcomes are all possible combinations.
• By calculating \( \left( \frac{1}{13,983,816} \right)^2 \), we determine the likelihood of winning twice in a row as an astronomically low probability.

Probability calculations help us quantify and understand the chances of various events, making them an essential tool in decision-making.