Problem 19 - 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 that this is the only way to win, what is the expected value of this game?

Step-by-Step Solution

Answer

The expected value of the lottery game is approximately \(-0.93\).

Step 1: Determine the Total Number of Combinations

In a 6/49 lottery, a player selects 6 numbers out of 49. The total number of combinations is calculated using the combination formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Here, \(n = 49\) and \(k = 6\): \[\binom{49}{6} = \frac{49!}{6! \cdot (49 - 6)!} = 13,983,816\]

Step 2: Calculate the Probability of Winning

The probability of selecting the winning numbers is the ratio of the favorable outcome to the total number of combinations. Since there is only one way to pick the six winning numbers, the probability \(P\) of winning is: \[P = \frac{1}{13,983,816}\]

Step 3: Calculate the Expected Value for Winning

The expected value from winning is the product of the win amount and the winning probability. The winning amount is \$1,000,000, so the expected value \(E_w\) is: \[E_w = 1,000,000 \times \frac{1}{13,983,816} = \frac{1,000,000}{13,983,816}\]

Step 4: Calculate the Expected Value from Playing

Since the cost to play the game is \\(1, the game incurs a cost or negative expected value of \\)1 for playing (because this is spent regardless of outcome).

Step 5: Calculate the Total Expected Value

The total expected value \(E\) is the expected value from winning minus the cost of playing the game:\[E = \frac{1,000,000}{13,983,816} - 1\]Converting \(\frac{1,000,000}{13,983,816}\) gives approximately \(0.0715\).Therefore, \[E = 0.0715 - 1 = -0.9285\]

Step 6: Interpret the Result

The negative expected value of \(-0.9285\) means that a player loses approximately \$0.93 per game on average. This signifies a long-term loss on playing the lottery.

Key Concepts

Lottery Probability

Lottery probability involves determining the chances of winning a game of chance. In a classic 6/49 lottery, you select 6 numbers from a pool of 49 numbers. The probability of winning is quite low because you must match your chosen numbers to the exact combination of randomly drawn numbers.

To find out how many possible combinations there are, you use the combination formula from combinatorics:
\[\binom{49}{6} = \frac{49!}{6!(49-6)!}\]
This calculates to 13,983,816 different possible combinations. This means there is only one chance in 13,983,816 of winning the jackpot. The probability of winning, therefore, is the number of winning combinations (just one!) divided by the total combinations:
\[ P = \frac{1}{13,983,816} \]

Understanding the probability:

• Extremely low due to high number of combinations.
• Reflects why it's difficult to win such lotteries.
Winning such games is rare due to the large pool of number combinations, which is why considering the odds is crucial before playing.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations, permutations, and the counting of outcomes in a structured way. It is particularly useful for solving problems related to probability, such as lottery games.

In the context of a 6/49 lottery, combinatorics lets us determine how many different ways you can select 6 numbers from a group of 49. This is calculated using the combination formula:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
Here, \(n = 49\) is the total number of different numbers to choose from, and \(k = 6\) is how many numbers you select. This yields 13,983,816 different combinations.

Key Points in Combinatorics:

• Factorials: Notated as \(!\), it means the product of an integer and all positive integers below it.
• Combination: Choosing items without regard to the order, which is crucial for calculating lottery probabilities.

Combinatorics breaks down complex problems by organizing outcomes into manageable groups, making it easier to compute probabilities in various contexts.

Probability Theory

Probability theory is the mathematical framework used to calculate the likelihood of various outcomes. It goes beyond counting possibilities, focusing on the likelihood each outcome actually happens.

In lotteries, probability theory helps explain the expected value, which is the average outcome if the lottery was played many times. Even if winning seems possible, probability theory shows the long-term gains or losses involved.

For lotteries, the expected value (EV) combines the win probability with the prize amount, while subtracting the cost to play:
\[ EV = (win \ amount \times win \ probability) - cost \ to \ play \]

Application of Probability Theory:

• Demonstrates the negative expectation of most lottery games.
• Explains why lotteries are profitable for operators over time.

Probability theory reveals that despite the temptation of huge prizes, the odds mean most players will lose money in the long run. This understanding can guide decisions on participation in lotteries.