Problem 20 - 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(all even-numbered seasons in the correct position)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{1680} \).

Step 1: Determine total possible arrangements

Since there are 8 DVD cases, each can be placed in one of 8 positions. Therefore, the total number of possible arrangements is given by the factorial of 8, which is:\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \]

Step 2: Identify cases to be positioned correctly

Identify the cases that are all even-numbered seasons — these correspond to seasons 2, 4, 6, and 8. We need these cases to be placed correctly in their respective positions.

Step 3: Calculate the number of favorable arrangements

For the favorable outcome, the cases of seasons 2, 4, 6, and 8 must remain in their respective positions. The rest of the DVD cases (seasons 1, 3, 5, and 7) can be arranged freely in the remaining 4 positions. The number of ways to arrange the 4 cases is the factorial of 4, thus:\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

Step 4: Compute the probability

The probability is the ratio of the number of favorable arrangements to the total number of possible arrangements, which is calculated as:\[ P = \frac{4!}{8!} = \frac{24}{40320} = \frac{1}{1680} \]

Key Concepts

Understanding Factorial

The concept of a factorial is an important one in probability and combinatorics. A factorial is the product of an integer and all the integers below it, down to one. It is expressed using an exclamation mark, like so: \( n! \). For example, the factorial of 3, written as \( 3! \), is calculated as \( 3 \times 2 \times 1 = 6 \).

Here's why factorials are so useful, especially in permutations and combinations:

• Factorials help calculate the total number of ways to arrange a set of objects.
• They are used to determine permutations, which are all the possible arrangements of a set of items.
• Factorials grow very quickly, so they are particularly useful for counting objects when the number of items gets large.

In the exercise, knowing there are 8 DVD cases, we use factorial to find all possible ways they could be arranged, \( 8! = 40320 \). This gives us a foundational understanding of the possible outcomes, preparing us for deeper insights into probability.

Permutations Explained

Permutations represent the different ways we can arrange a certain number of objects in sequence. When thinking of permutations, consider not just what is present, but also the order in which each element appears. Order matters greatly in permutations.

For example, in our exercise, even though seasons 2, 4, 6, and 8 must remain in their positions, the permutations of the remaining DVD cases truly count. We have four other DVD seasons (1, 3, 5, 7) that can be rearranged. These are calculated as \( 4! = 24 \) permutations.

To understand permutations further consider:

• A permutation is suitable when each arrangement of the set counts as a different outcome.
• If the set involves unique items or when specific order is crucial, permutations are used.
• Calculating permutations involves using factorials, as seen in \( n! \) for arrangements.

Recognizing when to see problems through a permutation lens can significantly guide us in solving complex problems like arranging items randomly.

Identifying Favorable Outcomes

In probability, favorable outcomes refer to the specific outcomes that satisfy the conditions of the given scenario. These are the particular results that we are actually interested in measuring or calculating.

In our exercise case: To reach the goal, where all even-numbered seasons are placed correctly, we need to focus on the arrangements where that specific condition is met.

Here's how to identify and count favorable outcomes:

• Firstly, define what the "favorable outcome" entails in the context.
• Determine the subset of arrangements that meet this defined criteria.
• For Janice’s DVDs, the favorable outcome occurs when seasons 2, 4, 6, and 8 remain untouched, which gives us \( 24 \) favorable permutations of the others.
• Calculating probability involves the ratio of favorable outcomes to all possible outcomes, such as \( \frac{1}{1680} \) in this case.

Understanding how to identify favorable outcomes aids us in calculating probabilities accurately, grounding our predictions and solutions in clear contexts.