Calculating the expected value (EV) of a random variable in a lottery game involves determining the average outcome if the game were played an infinite number of times. This helps to understand the long-term average result you can expect from participating in the lottery.
Steps to Calculate the Expected Value
1. Identify the Possible Outcomes: List all possible outcomes of the lottery game, including the prizes and the number of combinations that yield each prize.
2. Determine the Probability of Each Outcome: Calculate the probability of each possible outcome. This is done by dividing the number of ways an outcome can occur by the total number of possible outcomes.
3. Calculate the Expected Value: Multiply each outcome's prize by its probability and sum these values.
Example: Simple Lottery Game
Consider a simplified lottery game where you pick 6 numbers from 1 to 49, similar to many national lotteries. The game has the following prize structure:
- Jackpot: Match 6 numbers (prize = $10,000,000)
- Second Prize: Match 5 numbers (prize = $100,000)
- Third Prize: Match 4 numbers (prize = $1,000)
- Fourth Prize: Match 3 numbers (prize = $100)
- Fifth Prize: Match 2 numbers (prize = $10)
- No Prize: Match fewer than 2 numbers (prize = $0)
Step-by-Step Calculation
1. Calculate the Total Number of Combinations
The total number of ways to choose 6 numbers from 49 is:
\[ C(49, 6) = \frac{49!}{6!(49-6)!} = 13,983,816 \]
2. Calculate the Probability of Each Outcome
- Jackpot (6/6):
\[ P(\text{Jackpot}) = \frac{1}{13,983,816} \]
- Second Prize (5/6):
\[ P(\text{Second Prize}) = \frac{C(6, 5) \times C(43, 1)}{C(49, 6)} = \frac{6 \times 43}{13,983,816} = \frac{258}{13,983,816} = \frac{1}{54,201} \]
- Third Prize (4/6):
\[ P(\text{Third Prize}) = \frac{C(6, 4) \times C(43, 2)}{C(49, 6)} = \frac{15 \times 903}{13,983,816} = \frac{13,545}{13,983,816} = \frac{1}{1,033} \]
- Fourth Prize (3/6):
\[ P(\text{Fourth Prize}) = \frac{C(6, 3) \times C(43, 3)}{C(49, 6)} = \frac{20 \times 12,341}{13,983,816} = \frac{246,820}{13,983,816} = \frac{1}{57} \]
- Fifth Prize (2/6):
\[ P(\text{Fifth Prize}) = \frac{C(6, 2) \times C(43, 4)}{C(49, 6)} = \frac{15 \times 148,995}{13,983,816} = \frac{2,234,925}{13,983,816} = \frac{1}{6.25} \]
- No Prize:
\[ P(\text{No Prize}) = 1 - P(\text{Jackpot}) - P(\text{Second Prize}) - P(\text{Third Prize}) - P(\text{Fourth Prize}) - P(\text{Fifth Prize}) \]
3. Calculate the Expected Value
The EV is the sum of each prize multiplied by its probability:
\[ \text{EV} = (10,000,000 \times P(\text{Jackpot})) + (100,000 \times P(\text{Second Prize})) + (1,000 \times P(\text{Third Prize})) + (100 \times P(\text{Fourth Prize})) + (10 \times P(\text{Fifth Prize})) + (0 \times P(\text{No Prize})) \]
Substituting the probabilities:
\[ \text{EV} = 10,000,000 \times \frac{1}{13,983,816} + 100,000 \times \frac{1}{54,201} + 1,000 \times \frac{1}{1,033} + 100 \times \frac{1}{57} + 10 \times \frac{1}{6.25} \]
\[ \text{EV} = 0.714 + 1.844 + 0.968 + 1.754 + 1.6 \]
\[ \text{EV} = 6.88 \]
So, the expected value of a ticket in this simplified lottery game is approximately $6.88. If a ticket costs more than this amount, then, on average, you would expect to lose money in the long run.
Summary
- Identify all possible outcomes and their associated prizes.
- Calculate the probability of each outcome.
- Multiply each prize by its probability and sum these products to get the expected value.
This calculation gives you a theoretical average return on a lottery ticket, helping to understand the long-term profitability (or loss) of participating in the lottery.
Steps to Calculate the Expected Value
1. Identify the Possible Outcomes: List all possible outcomes of the lottery game, including the prizes and the number of combinations that yield each prize.
2. Determine the Probability of Each Outcome: Calculate the probability of each possible outcome. This is done by dividing the number of ways an outcome can occur by the total number of possible outcomes.
3. Calculate the Expected Value: Multiply each outcome's prize by its probability and sum these values.
Example: Simple Lottery Game
Consider a simplified lottery game where you pick 6 numbers from 1 to 49, similar to many national lotteries. The game has the following prize structure:
- Jackpot: Match 6 numbers (prize = $10,000,000)
- Second Prize: Match 5 numbers (prize = $100,000)
- Third Prize: Match 4 numbers (prize = $1,000)
- Fourth Prize: Match 3 numbers (prize = $100)
- Fifth Prize: Match 2 numbers (prize = $10)
- No Prize: Match fewer than 2 numbers (prize = $0)
Step-by-Step Calculation
1. Calculate the Total Number of Combinations
The total number of ways to choose 6 numbers from 49 is:
\[ C(49, 6) = \frac{49!}{6!(49-6)!} = 13,983,816 \]
2. Calculate the Probability of Each Outcome
- Jackpot (6/6):
\[ P(\text{Jackpot}) = \frac{1}{13,983,816} \]
- Second Prize (5/6):
\[ P(\text{Second Prize}) = \frac{C(6, 5) \times C(43, 1)}{C(49, 6)} = \frac{6 \times 43}{13,983,816} = \frac{258}{13,983,816} = \frac{1}{54,201} \]
- Third Prize (4/6):
\[ P(\text{Third Prize}) = \frac{C(6, 4) \times C(43, 2)}{C(49, 6)} = \frac{15 \times 903}{13,983,816} = \frac{13,545}{13,983,816} = \frac{1}{1,033} \]
- Fourth Prize (3/6):
\[ P(\text{Fourth Prize}) = \frac{C(6, 3) \times C(43, 3)}{C(49, 6)} = \frac{20 \times 12,341}{13,983,816} = \frac{246,820}{13,983,816} = \frac{1}{57} \]
- Fifth Prize (2/6):
\[ P(\text{Fifth Prize}) = \frac{C(6, 2) \times C(43, 4)}{C(49, 6)} = \frac{15 \times 148,995}{13,983,816} = \frac{2,234,925}{13,983,816} = \frac{1}{6.25} \]
- No Prize:
\[ P(\text{No Prize}) = 1 - P(\text{Jackpot}) - P(\text{Second Prize}) - P(\text{Third Prize}) - P(\text{Fourth Prize}) - P(\text{Fifth Prize}) \]
3. Calculate the Expected Value
The EV is the sum of each prize multiplied by its probability:
\[ \text{EV} = (10,000,000 \times P(\text{Jackpot})) + (100,000 \times P(\text{Second Prize})) + (1,000 \times P(\text{Third Prize})) + (100 \times P(\text{Fourth Prize})) + (10 \times P(\text{Fifth Prize})) + (0 \times P(\text{No Prize})) \]
Substituting the probabilities:
\[ \text{EV} = 10,000,000 \times \frac{1}{13,983,816} + 100,000 \times \frac{1}{54,201} + 1,000 \times \frac{1}{1,033} + 100 \times \frac{1}{57} + 10 \times \frac{1}{6.25} \]
\[ \text{EV} = 0.714 + 1.844 + 0.968 + 1.754 + 1.6 \]
\[ \text{EV} = 6.88 \]
So, the expected value of a ticket in this simplified lottery game is approximately $6.88. If a ticket costs more than this amount, then, on average, you would expect to lose money in the long run.
Summary
- Identify all possible outcomes and their associated prizes.
- Calculate the probability of each outcome.
- Multiply each prize by its probability and sum these products to get the expected value.
This calculation gives you a theoretical average return on a lottery ticket, helping to understand the long-term profitability (or loss) of participating in the lottery.
