Problem 52 - Algebra 2

Question

Tiles numbered from 1 to 6 are placed in a bag and are drawn out to determine which of six tasks will be assigned to six people. What is the probability that the tiles numbered 5 and 6 are drawn consecutively? F. \(\frac{2}{3}\) G. \(\frac{2}{5}\) H. \(\frac{1}{2}\) J. \(\frac{1}{3}\)

Step-by-Step Solution

Answer

The probability is \(\frac{1}{3}\), matching option J.

Step 1: Understand the Problem

We have 6 tiles numbered from 1 to 6, and we want to calculate the probability that tiles numbered 5 and 6 are drawn consecutively, in any order.

Step 2: Calculate Total Possible Arrangements

First, calculate the total number of possible arrangements of the 6 tiles. This can be done by determining the number of permutations of the 6 tiles, given by the factorial of 6: \[ 6! = 720 \]

Step 3: Determine Consecutive Situation

Next, treat tiles 5 and 6 as a pair. This means we are arranging 5 groups (1, 2, 3, 4, and the pair {5, 6}), and within the pair {5, 6}, we can arrange them in 2 different ways (5 then 6 or 6 then 5). So the number of desired arrangements is: \[ 5! \times 2 = 120 \times 2 = 240 \]

Step 4: Calculate Probability

The probability of 5 and 6 being consecutive is the ratio of favorable outcomes to total possible outcomes: \[ \text{Probability} = \frac{240}{720} = \frac{1}{3} \]

Step 5: Compare with Options

The calculated probability \(\frac{1}{3}\) matches option J. Therefore, J is the correct answer.

Key Concepts

Permutations

Permutations are a fundamental concept in probability and combinatorics. They involve arranging a set number of elements in various orders. For example, with 6 tiles numbered from 1 to 6, we can calculate all the possible sequences in which these tiles can appear. This process is crucial for solving problems where the order of elements matters.

• Order of arrangement is important; each different order counts as a separate permutation.
• To find all permutations of a set, we use factorial calculations.
• An example of permutations is arranging tiles to assign tasks, where each potential sequence of tasks is a different permutation.

Permutations help us determine the complete set of possible outcomes when order cannot be ignored.

Factorial

The concept of a factorial is central to calculating permutations. Factorial, represented by the symbol \(!\), is the product of all positive integers up to a given number. For instance, the factorial of 6, written as \(6!\), is calculated as follows:

• \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\)

Factorials are used to find the total number of ways to arrange a set of objects. In this exercise:

• We calculate \(6!\) to determine the total possible arrangements of the tiles.
• Factorials provide a straightforward method to cover all permutations.

Understanding factorials simplifies the process of working with permutations and is a key tool in probability calculations.

Favorable Outcomes

In probability, a favorable outcome is the scenario you want to count when determining the probability of an event. To find out how likely an event is, we determine favorable outcomes and compare them to all possible outcomes. In this exercise:

• We want to know how often tiles 5 and 6 are drawn consecutively.
• By treating 5 and 6 as a "block" or single unit, we simplify our arrangement process.
• The favorable outcomes are calculated by determining how many valid arrangements include the block {5, 6} or {6, 5}.

Being able to identify favorable outcomes is vital for calculating accurate probabilities. It involves a clear understanding of what constitutes success in a given problem.

Consecutive Numbers

Consecutive numbers are numbers that follow each other in order without any gaps. In this problem, consecutive numbers are crucial to finding the solution. We are specifically interested in when the numbers 5 and 6 appear next to each other in a sequence:

• We treat these consecutive numbers as a single unit of numbers to be arranged.
• This simplifies our problem because instead of treating them as 2 separate entities, we cluster them together.
• The number of ways to arrange these consecutive numbers, remaining close as a block, reduces the problem to arranging fewer elements.

Understanding how to group consecutive numbers can make complex arrangements more manageable, leading to easier calculations of probability.