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Many states have a lottery game, usually called a Pick-4, in which you pick a four-digit number such as 7359 . During the lottery drawing, there are four bins, each containing balls numbered 0 through 9\. One ball is drawn from each bin to form the four-digit winning number. a. You purchase one ticket with one four-digit number. What is the probability that you will win this lottery game? b. There are many variations of this game. The primary variation allows you to win if the four digits in your number are selected in any order as long as they are the same four digits as obtained by the lottery agency. For example, if you pick four digits making the number 1265, then you will win if \(1265,2615,5216,6521\), and so forth, are drawn. The variations of the lottery game depend on how many unique digits are in your number. Consider the following four different versions of this game. i. All four digits are unique (e.g., 1234 ) ii. Exactly one of the digits appears twice (e.g., 1223 or 9095 ) iii. Two digits each appear twice (e.g., 2121 or 5588 ) iv. One digit appears three times (e.g., 3335 or 2722 ) Find the probability that you will win this lottery in each of these four situations.

Step by step solution

01

Calculating the Probability of Winning the straight Lottery Game

First, it's necessary to understand that each of the four spots (units, tens, hundreds, thousands) for this four-digit number can be any of the 10 digits, 0 through 9. So, the total number of possible outcomes is \(10^4\) or 10,000. Since your ticket is a single four-digit number, the desired outcome is just ONE (your chosen number). Therefore, the probability of winning the lottery game is: \(1/10000\) or 0.0001.

02

Probability in Different Variations

For all four unique digits, you will win the lottery if your four digits are drawn in any order. There are 4!=24 permutations for your four digits. So, the probability of winning using a number with four unique digits is: \(24/10000\) or 0.0024. For exactly one digit appearing twice, there are \(4P2 = 12\) permutations for these digits. Probability of winning is: \(12/10000\) or 0.0012. For the case of two digits each appearing twice, there are 6 possible permutations for these four digits. So, the probability of winning is: \(6/10000\) or 0.0006. Finally, if one digit appears three times, there are 4 possible arrangements of that number. So, the probability of your winning is: \(4/10000\) or 0.0004.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Permutation

Imagine you have a set of items, and you want to determine how many different ways you can arrange them. This is what permutations are all about. In our lottery example, the arrangement of four digits in different orders is a type of permutation. Each unique arrangement of these digits represents a different permutation.

A key principle in permutations is that the order of the items matters. This means that 1234 and 4321 are considered different permutations because the sequence is not the same. The formula for finding permutations when all items are unique is represented by factorial notation and written as: \[ n! = n \times (n-1) \times (n-2) \times ... \times 1 \]Here, \(n!\) denotes the number of ways to arrange \(n\) items.

Lottery game

A lottery game is a type of gambling where participants choose numbers in hopes of matching those drawn by a lottery organization. These games are all about chance, and their results rely solely on probability. Our Pick-4 lottery game requires participants to choose a four-digit number from 0000 to 9999. The possibilities are vast—10,000 different outcomes! This is because each digit can independently be anything from 0 to 9, making each possible draw unique.

Winning in this lottery isn't easy. The probability of holding the winning combination in the exact sequence drawn (known as a 'straight' win) is slim—a mere \(1/10000\) or 0.0001. However, variations increase the chances slightly by allowing wins if numbers are correct but not in the exact order. Understanding lottery games helps to grasp basic probability concepts and the randomness inherent in these types of games.

Combinatorics

Combinatorics is a mathematical field that studies the counting and arrangement of objects. It's crucial for solving problems where you need to manage or group items. In our lottery game, combinatorics helps us solve how many different ways we can order the digits or group digits when some are repeated.

In this particular game, combinatorics principles are applied when calculating outcomes with varying numbers of unique digits. For instance:

• With all four unique digits, you can reshuffle the numbers any of 24 ways using 4! permutations.
• If exactly one digit repeats, the formula adjusted to account for repeated digits is used: \(\frac{n!}{k!}\), where \(n\) is total digits and \(k\) is repeated digits.

Combinatorics provides the tools needed to calculate permutations and combinations, revealing the wondrous ways items can be organized or grouped.

Unique digits

Unique digits refer to using distinct numbers without repetition within a set. In the context of our lottery example, it means choosing digits like 1234 or 5678, where no number repeats. When all digits are unique in a Pick-4 lottery, there are more permutations because all figures can be freely rearranged without running into duplicates.

For example, 4 unique digits can be rearranged in \(4!\) or 24 different ways. But, uniqueness isn't just interesting for rearrangement potential; it plays a significant role in predicting probabilities. It impacts the variations within the game, altering the odds of drawing that particular arrangement when compared to sets with repeating digits. Understanding unique digits helps in appreciating the complexity and nuances of probability in every draw.