Problem 17 - Algebra and Trigonometry

Question

Lottery In a 6\(/ 49\) lottery game, a player pays \(\$ 1\) and selects six numbers from 1 to \(49 .\) Any player who has chosen the six winning numbers wins \(\$ 1,000,000\) . Assuming this is the only way to win, what is the expected value of this game?

Step-by-Step Solution

Answer

The expected value of playing the lottery is approximately -$0.93.

Step 1: Understanding the Lottery Mechanics

In a 6/49 lottery, the player chooses 6 numbers from a set of 49. The order of the numbers doesn't matter, and the player wins $1,000,000 if all 6 numbers match the drawn numbers exactly.

Step 2: Calculate the Total Number of Combinations

The total number of ways to choose 6 numbers from 49 is calculated using the combination formula \( \binom{n}{k} \), where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose. Thus, the number of combinations is: \( \binom{49}{6} = \frac{49!}{6!(49-6)!} \). Calculating this gives us \( 13,983,816 \) combinations.

Step 3: Determine the Probability of Winning

The probability of winning is the number of winning combinations (1) divided by the total number of combinations. Therefore, the probability is \( \frac{1}{13,983,816} \).

Step 4: Calculate the Expected Value of Winnings

The expected value is calculated by multiplying the probability of each outcome by its respective winnings. The only winning outcome gives $1,000,000 with a probability of \( \frac{1}{13,983,816} \). Hence, the expected value from winning is \( \frac{1}{13,983,816} \times 1,000,000 \approx 0.0715 \).

Step 5: Calculate the Expected Value Including the Cost of the Ticket

The expected loss from the ticket cost must also be included in the expected value calculation. The player spends $1 regardless of the outcome. Therefore, the expected value of this scenario becomes: \( 0.0715 - 1 = -0.9285 \).

Key Concepts

Combination Formula

In probability and statistics, the combination formula helps us determine how many ways we can select a group of items from a larger set, where the order does not matter. This is important when dealing with lottery games like the 6/49 lottery. To understand combinations, think of how to choose 6 numbers out of 49.

The formula for combinations is given by:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Where:

• \( n \) is the total number of items to choose from.
• \( k \) is the number of items to be chosen.

For the 6/49 lottery, \( n \) is 49, and \( k \) is 6. Using this, the number of combinations is \( \binom{49}{6} = 13,983,816 \). This number means there are almost 14 million ways to choose 6 numbers from a set of 49.

Understanding combinations is crucial as it forms the basis for calculating the probability of winning.

Probability of Winning

Probability is a measure of how likely an event is to occur. In the context of lottery games, it refers to the chances of selecting the correct numbers. For the 6/49 lottery, the probability of winning can be determined once we know the total number of possible combinations.

Since there is only one set of winning numbers, the probability of selecting these exact numbers is:

• \( \frac{1}{13,983,816} \)

This probability is incredibly low, reflecting the difficulty of winning such lottery games.

The probability is vital as it influences the decision-making process of players, by giving a clear picture of how feasible winning really is.

Lottery Mathematics

Lottery mathematics involves understanding the various probabilities and outcomes involved in lottery games. Using mathematical principles, we can assess the risks and rewards of playing the lottery.

Key concepts include:

• Combination Calculations: Determine the total possible selections.
• Probability Calculations: Assess the likelihood of selecting the winning numbers.
• Expected Value Calculations: Understand the financial implications of playing.

By applying these principles, we can make more informed decisions about whether playing a lottery game is worth the investment.

Lottery mathematics helps us break down the complexity of games of chance into understandable components using mathematical tools.

Mathematical Expectation

Mathematical expectation, or expected value, is a key concept in probability that refers to the average outcome we would expect from a random event if it were repeated many times. For lotteries, it gives an idea of the average winnings, including losses, over the long term.

The expected value for the 6/49 lottery is calculated by:

• Multiplying the probability of a win by the prize amount, \( 1,000,000 \).
• Subtracting the cost of the ticket, typically \( 1 \).

The result here is:

• \( \frac{1}{13,983,816} \times 1,000,000 = 0.0715 \)
• The expected value becomes \( 0.0715 - 1 = -0.9285 \)

This negative expected value means that, on average, a player would lose money every time they buy a ticket. Thus, mathematical expectation helps visualize the lottery as a game where the odds of financial gain are not in favor of the player.