Problem 21 - Algebra 2

Question

Janice has 8 DVD cases on a shelf, one for each season of her favorite TV show. Her brother accidentally knocks them off the shelf onto the floor. When her brother puts them back on the shelf, he does not pay attention to the season numbers and puts the cases back on the shelf randomly. Find each probability. P(seasons 5 through 8 in any order followed by seasons 1 through 4 in any order)

Step-by-Step Solution

Answer

The probability is \(\frac{1}{70}\).

Step 1: Total Arrangements

To find the probability, first determine the total number of ways to arrange 8 DVD cases. The total number of arrangements can be calculated using the factorial of the number of cases. Thus, the total arrangements is given by \(8!\), which is the product of all positive integers up to 8: \(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

Step 2: Specific Arrangement Requirement

Next, determine the number of ways to arrange seasons 5 through 8 in any order followed by seasons 1 through 4 in any order. First, arrange seasons 5 to 8, which can be done in \(4!\) ways. Then, arrange seasons 1 to 4, which again can be done in \(4!\) ways. So the specific arrangement is \(4! \times 4!\).

Step 3: Calculate the Probability

The probability is the number of favorable specific arrangements divided by the total arrangements. Use the formula: \( \frac{4! \times 4!}{8!} \). Calculate each component: \(4! = 24\) and \(8! = 40320\). Substitute these values into the equation: \( \frac{24 \times 24}{40320} = \frac{576}{40320}\). Simplify this fraction to \( \frac{1}{70} \) by dividing the numerator and the denominator by their greatest common divisor.

Key Concepts

Factorials

Factorials are a way to multiply a series of descending natural numbers. They are used to calculate permutations and combinations of objects. The factorial of a number, shown as \( n! \), means you multiply the number by every whole number less than itself down to 1. For example, \( 5! \) is calculated as \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Factorials grow quite rapidly with increased numbers. For instance, \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320 \). Each addition to the factorial sequence makes the total much larger.
Understanding how to calculate and simplify factorials is essential in solving problems in probability, especially when determining possible permutations or combinations.

Combinatorics

Combinatorics deals with counting, arranging, and finding patterns in a set. It is a mathematical study of counting things. This field includes concepts such as permutations and combinations that tell you how many ways things can be arranged or combined.
In this exercise, combinatorics helps us figure out all the possible ways eight DVD cases can be arranged. This procedural arrangement helps us calculate probabilities by identifying the amount of desired orderings versus total orderings.

• Using combinatorics, we analyze possible arrangements, considering each choice one step at a time. This involves computing possible favorable outcomes (like certain DVDs being next to each other) over total possibilities.
• It opens up applications in different fields like computer science, logistics, and even romance – like determining how many ways you can relay a message using a limited set of phrases. If you're planning or optimizing tasks, combinatorics gives a structured strategy.

Permutations

Permutations are specific arrangements of a set of objects where order is important. For example, having three letters A, B, C will result in permutations such as ABC, ACB, BAC, and so on.
In our exercise, we're interested in the permutation of DVD seasons, as each arrangement of 8 DVDs gives a unique sequence. Calculating the permutation of seasons 5 through 8 being followed by seasons 1 through 4 involves finding the number of ways to arrange four items, which is \(4!\). This yields \(4! = 24\) permutations.
Understanding permutations is vital in scenarios where the sequence of arrangement matters, such as scheduling, DNA sequencing, and seating arrangements in events.

Probability Calculation

Probability calculation is about figuring out how likely an event is to happen. This involves a division of favorable outcomes over total possible outcomes.
In our problem, the probability we need is the chance of a specific sequence of DVDs order occurring, which is calculated as:

• The number of favorable arrangements (where seasons 5-8 come first, followed by seasons 1-4) is given by multiplying the permutations of those two sections: \(4! \times 4! = 576\).
• The total possible arrangements for the DVDs are the permutations of all eight cases: \(8! = 40,320\).

Thus, the probability is \(\frac{576}{40,320} = \frac{1}{70}\), showing that this particular arrangement is quite unlikely.
Probability calculations are crucial in assessing risk, decision-making, and predictions in games, weather forecasts, and everyday choices like deciding whether to carry an umbrella.