Question

Beat the lottery. Many states run lotteries to raise money. A Web site advertises that it knows "how to increase YOUR chances of Winning the Lottery." They offer several systems and criticize others as foolish. One system is called Lucky Numbers. People who play the Lucky Numbers system just pick a "lucky" number to play, but maybe some numbers are luckier than others. Let's use a simulation to see how well this system works. To make the situation manageable, simulate a simple lottery in which a single digit from 0 to 9 is selected as the winning number. Pick a single value to bet, such as 1 , and keep playing it over and over. You'll want to run at least 100 trials. (If you can program the simulations on a computer, run several hundred. Or generalize the questions to a lottery that chooses two- or three-digit numbers-for which you'll need thousands of trials.) a) What proportion of the time do you expect to win? b) Would you expect better results if you picked a "luckier" number, such as \(7 ?\) (Try it if you don't know.) Explain.

Short answer

The win probability is about 10%, independent of the chosen number.

Step-by-step solution

Understand the Problem

The problem involves simulating a simple lottery to determine the effectiveness of picking a 'lucky' number repeatedly. We'll simulate the selection of a single-digit number (0 to 9) and calculate the win proportion over many trials.

Set Up the Simulation

Choose a fixed lucky number to bet on, such as 1. We will simulate at least 100 trials where a winning number is randomly chosen from 0 to 9, and check how many times the chosen lucky number (1 in this case) is the winning number.

Calculate Expected Proportion

In a fair lottery system, each number from 0 to 9 has an equal chance of being chosen. Thus, the probability of any single number being the winning number is \(\frac{1}{10}\) or 10%.

Run the Simulation

Using either a computer program or manually using a random number generator, simulate 100 trials where a random digit is chosen for each trial. Count the number of times the chosen number (e.g., 1) matches the winning number.

Analyze the Results

Calculate the proportion of trials where the chosen number wins. This should be close to the expected theoretical probability of 10%, assuming random number generation is fair.

Consider Different Numbers

Try running the simulation using a different chosen number (such as 7) and verify if the win proportion changes significantly. Since all numbers have an equal chance, this proportion should remain close to 10% regardless of the number chosen.

Key concepts

Lottery Probability

In a lottery, you're guessing the outcome of a random event. Each number you can bet on has the same chance of being selected. Imagine picking a single digit from 0 to 9. That's a basic lottery with only ten possible outcomes. When you pick a number to bet on, like '1,' the probability of winning is determined by the number of possible winning outcomes. In our simple lottery, there are 10 outcomes, hence your chance to pick the winning number is

• 1 in 10, represented as a probability of \( \frac{1}{10} \).
• This translates to a probability of 0.1 or 10%.

No number is inherently luckier than the others. Each attempt is an independent event with the same 10% winning probability. Therefore, the belief in luckier numbers is a matter of superstition rather than statistical likelihood.

Random Number Generation

Random number generation is vital in simulating probabilities like those in a lottery. It allows us to imitate the process of randomly selecting a digit from 0 to 9. This can be done using:

• Physical means, like drawing numbers from a hat.
• Or computer-based generators, which mimic randomness.

These generators produce sequences of numbers that appear random. Each number in the sequence is selected without any pattern, making it perfect for lottery simulations. Importantly, random number generators ensure each digit, from 0 to 9, has an equal chance of selection in each draw. For example, in our 10-digit lottery simulation:

• A random digit picked in each trial represents the winning number.
• By running these trials hundreds of times, we can observe the law of large numbers in action, where the results approach the theoretical probability of 10% for each number.

Expected Value Calculation

Expected value in probability is a way to anticipate the average outcome. It calculates what you would expect to win if you could play an infinite number of times. For our single-digit lottery, this is how it's done:

• Each round has a win probability of \( \frac{1}{10} \), hence a loss probability of \( \frac{9}{10} \).
• If you bet 1 currency unit and win, you gain back your 1 unit plus a prize.

To estimate expected value, consider the simplified scenario where the prize equals your bet (1 unit):

• The expected value E for winning is \( 1 \times \frac{1}{10} \).
• Subtract your usual bet, which you lose if you don’t win: \( - 1 \times \frac{9}{10} \).
• The total expected value becomes \( \frac{1}{10} - \frac{9}{10} = -0.8 \).

This negative expected value suggests that over many plays, you are more likely to lose than win, typical for games of chance like lotteries. It indicates you shouldn’t expect to make money from the lottery overall unless the prize greatly outweighs the losses.