Question

On September 11,2002, the first anniversary of the terrorist attack on the World Trade Center, the New York State Lottery's daily number came up \(9-1-1 .\) An interesting coincidence or a cosmic sign? a) What is the probability that the winning three numbers match the date on any given day? b) What is the probability that a whole year passes without this happening? c) What is the probability that the date and winning lottery number match at least once during any year? d) If every one of the 50 states has a three-digit lottery, what is the probability that at least one of them will come up \(9-1-1\) on September \(11 ?\)

Short answer

a) 0.001 b) 0.714 c) 0.286 d) 0.049

Step-by-step solution

Calculate total possible outcomes

In a daily three-digit lottery, each number is independently chosen from 0 to 9. Therefore, the total number of different possible outcomes is given by multiplying the number of choices for each digit: 10 choices per digit, and 3 digits total. Thus, there are \[ \text{Total outcomes} = 10 \times 10 \times 10 = 1000. \]

Calculate favorable outcomes for matching the date

The favorable outcome for the date matching is the specific sequence '9-1-1'. There is only one such sequence possible. Thus, we have:\[ \text{Favorable outcomes} = 1. \]

Calculate probability of matching on a given day

The probability that the lottery number matches the date 9-1-1 on any given day is the ratio of the number of favorable outcomes to the total number of outcomes:\[ P(\text{match on a day}) = \frac{1}{1000}. \]

Calculate probability of no match over a year

We assume the lottery is drawn every day, so there are 365 independent draws in a year. The probability of not matching on a given day is\[ P(\text{no match}) = 1 - P(\text{match}) = 1 - \frac{1}{1000} = 0.999. \]The probability that this non-match condition holds for an entire year is\[ P(\text{no match in a year}) = (0.999)^{365}. \]

Calculate probability of at least one match during a year

The probability of at least one match during a year is the complement of the probability of no matches over the year:\[ P(\text{at least one match in a year}) = 1 - P(\text{no match in a year}) = 1 - (0.999)^{365}. \]

Calculate probability of at least one match in 50 states

Considering all 50 states have independent lotteries on September 11, the probability that at least one state's result matches '9-1-1' is the complement of the probability that none of the states' results match:\[ P(\text{no state matches}) = \left( 1 - \frac{1}{1000} \right)^{50} \approx 0.951. \]Thus, the probability that at least one state's result is '9-1-1' is:\[ P(\text{at least one state matches}) = 1 - 0.951. \]

Key concepts

Lottery Probability

Lottery probability is the likelihood of a particular outcome occurring in a lottery draw. In the case of a three-digit lottery game, each digit can take any value from 0 to 9, resulting in 10 possible options for each digit. This gives us a total number of possible outcomes calculated using the formula:\[ \text{Total outcomes} = 10 \times 10 \times 10 = 1000 \]This means that there are 1000 different sequences that can be the winning three-digit number each day. When calculating the probability of drawing a specific sequence like "9-1-1," we look at the number of favorable outcomes, which in this case is 1—the exact sequence "9-1-1." Thus, the probability is the ratio of favorable outcomes to total outcomes:\[ P(\text{match on a day}) = \frac{1}{1000} \]Understanding these outcomes allows us to predict the likelihood of such numbers appearing in a lottery draw, emphasizing the random nature and fairness of lottery-based systems.

Complement Rule

The complement rule is a fundamental principle in probability, which is particularly useful in calculating the likelihood of an event occurring at least once. It states that the probability of an event not happening is 1 minus the probability of it happening. For instance, if the probability of winning a jackpot is \( \frac{1}{1000} \), the probability of not winning is:\[ P(\text{no match}) = 1 - \frac{1}{1000} = 0.999 \]When applied over multiple trials, like a year (365 days), we use the power of the complement to ascertain that non-occurrence over repeated attempts. We calculate this by raising the probability of a single non-match to the power of the number of trials:\[ P(\text{no match in a year}) = (0.999)^{365} \]Therefore, the probability of at least one match in a year is the complement of all non-matches:\[ P(\text{at least one match in a year}) = 1 - (0.999)^{365} \]This approach is particularly useful for events that are rare but have multiple opportunities to occur, like winning with a specific lottery number.

Independent Events

Independent events in probability are such that the outcome of one event does not affect the outcome of another. This principle is crucial in lottery games, as each draw is independent of others. In our exercise, each state's lottery is considered an independent event.For example, on September 11, each state has its lottery draw, and the outcome in one state does not influence the other states’ outcomes. This means that the occurrence probability in multiple states remains consistent across each one:\[ P(\text{no match in a state}) = \left( 1 - \frac{1}{1000} \right) \]Thus, for all 50 states, the calculations remain multiplicative, reflecting independence:\[ P(\text{no state matches}) = (0.999)^{50} \approx 0.951 \]Given this, the probability that at least one state's result is "9-1-1" is:\[ P(\text{at least one state matches}) = 1 - 0.951 \]This independent event calculation helps demonstrate how probabilities multiply when considering multi-event scenarios.

Random Outcomes

In probability, a random outcome is one that is unpredictable and selected purely by chance. Each draw in a lottery is a random event, meaning that any sequence of numbers is just as likely to occur as any other. The probability of drawing any specific number, such as "9-1-1," relies purely on randomness rather than patterns or past outcomes. Here are the key points to note about random outcomes:

• Each outcome is equally likely.
• Previous results do not affect future draws.
• The randomness ensures fairness in lottery systems.

This property makes lottery games exciting and unpredictable. Since each number is chosen randomly, even moments that seem significant, like drawing "9-1-1" on the anniversary of 9/11, are coincidences from a probability perspective. Recognizing the role of random outcomes helps students understand the unbiased nature of such events and appreciate their unpredictability.