Problem 66 - Basic Statistics for Business and Economics with Student

Question

For the daily lottery game in Illinois, participants select three numbers between 0 and \(9 .\) A number cannot be selected more than once, so a winning ticket could be, say, 307 but not 337 . Purchasing one ticket allows you to select one set of numbers. The winning numbers are announced on TV each night. a. How many different outcomes (three-digit numbers) are possible? b. If you purchase a ticket for the game tonight, what is the likelihood you will win? c. Suppose you purchase three tickets for tonight's drawing and select a different number for each ticket. What is the probability that you will not win with any of the tickets?

Step-by-Step Solution

Answer

a. 720 outcomes b. \( \frac{1}{720} \) c. Approximately 0.996

Step 1: Calculate Possible Outcomes

Since each digit can be a number from 0 to 9, the first digit has 10 possibilities. However, the second digit cannot be the first number, leaving 9 choices, and the third digit cannot be either of the first two digits, leaving 8 choices. Calculate the total number of distinct three-digit numbers by multiplying these possibilities: \(10 \times 9 \times 8\). This equals 720.

Step 2: Calculate Probability of Winning with One Ticket

The probability of winning with a single ticket is the ratio of winning outcomes to the total possible outcomes. Since there is only one winning combination, this probability is \( \frac{1}{720} \).

Step 3: Calculate Probability of Losing with One Ticket

To find the probability of not winning with one ticket, calculate the complement of winning with a ticket: \( 1 - \frac{1}{720} = \frac{719}{720} \).

Step 4: Calculate Probability of Losing with Three Tickets

Calculate the probability of losing with all three tickets. Each ticket is an independent event, so the probability of all losing is the product of losing with each ticket: \( \left( \frac{719}{720} \right)^3 \). Calculate this value to be approximately 0.996.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics centered around counting, arrangement, and combination of objects. It helps us answer questions like how many different possibilities exist when we choose or arrange certain items.
In the context of our lottery problem, combinatorics is used to determine the number of different three-digit outcomes when selecting each digit from 0 to 9 without repetition. This is essentially a permutation problem, where the order of numbers matters.
Here is how we calculate it:

• The first digit can be any number from 0 to 9, giving us 10 choices.
• Once the first digit is chosen, the second digit can be any number except the first, resulting in 9 choices.
• The third digit then has 8 choices, excluding the first two selected numbers.

Finally, we multiply these choices: \(10 \times 9 \times 8 = 720\). This tells us there are 720 unique three-digit combinations possible under these conditions.

Lottery Mathematics

Lottery mathematics often involves understanding the chances of winning a game, which is the probability of selecting the correct numbers.
To find the probability of winning this Illinois lottery, we need to know both the number of successful outcomes and the total possible outcomes.

• In a simple lottery, like this one, there is exactly one winning combination each time.
• Since we calculated 720 possible outcomes using combinatorics, the probability of choosing the correct set with one ticket is \(\frac{1}{720}\).
• This illustrates how unlikely it is to win with just a single ticket.

Each ticket you buy represents just one possible outcome. Therefore, if you purchase three tickets, each with different numbers, your chance of not winning with any of them is the complement of winning probability, calculated independently for each ticket. With three tickets, this probability becomes \(\left(\frac{719}{720}\right)^3\), a high value indicating a low chance of winning even with multiple entries.

Statistical Analysis

Statistical analysis helps us interpret probabilities and outcomes in various scenarios. It allows us to make informed predictions based on available data.
In the lottery exercise, we used statistical tools to determine the likelihood of losing with multiple tickets. Each selection of numbers is an independent event, meaning the outcome of one does not influence another.

• We calculated the losing probability for a single ticket: \(1 - \frac{1}{720} = \frac{719}{720}\).
• Then for multiple tickets, we find the probability of all losing by raising the losing probability to the number of tickets: \(\left(\frac{719}{720}\right)^3\).
• This yields a probability of approximately 0.996. Thus, statistically, there's a 99.6% chance that none of the three tickets win, demonstrating how rare winning can be.

Understanding these probabilities can help us make better decisions, like how many tickets might be a reasonable purchase, knowing the slim chances of winning.