Problem 42 - Algebra and Trigonometry

Question

In the 6/49 lottery game a player selects six numbers from 1 to 49. What is the probability of selecting at least five of the six winning numbers?

Step-by-Step Solution

Answer

The probability of selecting at least five winning numbers is approximately 0.0000185.

Step 1: Determine Possible Outcomes

First, we need to calculate the total number of possible outcomes for selecting 6 numbers from 49. This can be calculated using combinations, which is given as \( \binom{n}{k} \), where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose. Thus, \( \binom{49}{6} \) gives us the total number of combinations:\[ \binom{49}{6} = \frac{49 \times 48 \times 47 \times 46 \times 45 \times 44}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 13,983,816 \]

Step 2: Calculate Successful Outcomes for 6 Winning Numbers

We want to determine how many ways we can select all six winning numbers. If all six numbers have to match, there is exactly 1 successful outcome, since there is only one way to pick the correct six numbers.

Step 3: Calculate Successful Outcomes for 5 Winning Numbers

Now, we need to determine the number of ways to select exactly five winning numbers. We can choose 5 correct numbers in \( \binom{6}{5} \) ways. There is also \( \binom{43}{1} \) way to choose the incorrect number from the remaining 43 numbers.\[ \binom{6}{5} \times \binom{43}{1} = 6 \times 43 = 258 \]

Step 4: Sum of Successful Outcomes

Add the successful outcomes from getting exactly 5 and 6 winning numbers:\[ 1 + 258 = 259 \]

Step 5: Probability Calculation

The probability is the ratio of the successful outcomes to the total possible outcomes. Calculate:\[ \text{Probability} = \frac{259}{13,983,816} \approx 0.0000185 \]

Key Concepts

Combinations

In mathematics, combinations are a way to determine how many ways you can choose a subset of items from a larger set, without worrying about the order. For example, when you choose 6 numbers from a total of 49 in a lottery game, each group of 6 is considered a combination. The formula used for combinations is \( \binom{n}{k} \), which calculates the number of ways to choose \( k \) items from \( n \) items:

• \( n \) is the total number of items to choose from, in this case, 49.
• \( k \) is the number of items to be chosen, in this case, 6.

Therefore, in the lottery, the number of possible combinations to choose 6 numbers from 49 is \( \binom{49}{6} = 13,983,816 \). This helps understand how vast the number of possible ticket combinations can be.

Lottery Mathematics

Lottery mathematics involves probability and statistics to understand the likelihood of winning in lottery-style games. The unique aspect of these games is that they rely on random selection of numbers. Despite the huge number of tickets possible, calculating the odds can give you a mathematical understanding of your chances to win. For a 6/49 lottery game, you select any 6 numbers out of 49, creating a unique combination. From a mathematical viewpoint, every set of numbers has the same probability of being chosen since it is random. Knowing how to calculate combinations aids in understanding just how rare winning a lottery can be, bringing insight into the lottery's randomness and the small chance of hitting the jackpot.

Successful Outcomes

A successful outcome in a lottery can refer to any ticket where the numbers match the winning draw. Depending on the game rules, different levels of successes may exist. In our example:

• The highest success, selecting all 6 numbers correctly, gives only \( 1 \) way—as there is only one winning combination out of 13,983,816 possible ones.
• Selecting 5 out of the 6 winning numbers is also a successful outcome, calculated a little differently. You pick any 5 out of 6 correct numbers: \( \binom{6}{5} = 6 \) ways. Then choose \( 1 \) incorrect number from the remaining 43, which is \( \binom{43}{1} = 43 \), leading to a total of \( 258 \) successful ways.

These successful combinations are crucial in computing the probability of winning, giving clarity on how many different ways exist to achieve success.

Probability Calculation

Probability calculation helps us understand the likelihood or chance of an event occurring. In lottery games, it's the ratio of successful outcomes to possible outcomes. With our example of a 6/49 lottery, the steps are:

• First, find the combination count, \( \binom{49}{6} = 13,983,816 \), representing all possible ways to select the numbers.
• Then, count the successful outcomes, where you matched either 5 or 6 numbers, totaling \( 1 \) (for all 6 matched) plus \( 258 \) (for 5 matches plus 1 incorrect) = \( 259 \) successful outcomes.
• Finally, divide the successful outcomes by the total possible ones to find the probability: \( \frac{259}{13,983,816} \approx 0.0000185 \).

Understanding this probability shows just how rare these lottery events are, highlighting the power of mathematical probability in real-world scenarios.