Problem 48 - Algebra for College Students

Question

63 Each of three letters is to be mailed in any one of five different mailboxes. What is the probability that all will be mailed in the same mailbox?

Step-by-Step Solution

Answer

The probability is \( \frac{1}{25} \).

Step 1: Determine Total Mailing Combinations

First, we need to determine how many total ways there are to mail the three letters. Each letter can be placed in any of the five mailboxes. Therefore, for each letter, there are 5 choices. For three letters, this results in a total of \[5 \times 5 \times 5 = 5^3 = 125\]total possible mailing combinations.

Step 2: Find Favorable Outcomes

Next, we identify the favorable outcomes where all letters go into the same mailbox. Each letter must be mailed to the same one of the five mailboxes. This results in 5 favorable outcomes, one for each mailbox.

Step 3: Calculate the Probability

The probability that all letters will be mailed in the same mailbox is the number of favorable outcomes divided by the total number of outcomes. Using the numbers from the previous steps, the probability is given by\[\frac{5}{125} = \frac{1}{25}\]

Key Concepts

Mailing Combinations

When dealing with problems involving combinations, it's important to understand how many different ways an action can be performed. This problem involves mailing three letters, where each letter can be deposited into any one of five mailboxes. To find the total number of mailing combinations, you think about each letter separately:

• The first letter has 5 options for mailboxes.
• The second letter also has 5 options, regardless of where the first letter went.
• Similarly, the third letter still has 5 options.This gives you a total of \[ 5 \times 5 \times 5 = 5^3 = 125 \]combinations. Each mailbox essentially "multiplies" the options available for the letters that follow, leading to this exponential growth in possibilities. By counting all the combinations, you ensure that you consider every possible way the letters could be mailed.

Favorable Outcomes

In probability, a favorable outcome is the scenario that matches the event you are interested in. Here, the event is that all three letters are mailed in the same postbox. Since there are five mailboxes, and all letters must go in one of these for the specific event to occur, each mailbox offers one favorable outcome:

• All letters in Mailbox 1
• All letters in Mailbox 2
• All letters in Mailbox 3
• All letters in Mailbox 4
• All letters in Mailbox 5

These are the only scenarios where the event "all letters in the same mailbox" happens. Thus, there are exactly 5 such favorable outcomes. Keeping track of what counts as a favorable outcome is crucial for determining probability.

Calculation Steps

Calculating the probability requires relating favorable outcomes to total possible outcomes. This relationship is the cornerstone of probability and is expressed mathematically by the formula:Probability = \( \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \)Given the exercise:

• Total possible outcomes: 125
• Favorable outcomes (same mailbox): 5

Thus, the probability that all letters will be mailed into the same mailbox is:\[ \frac{5}{125} = \frac{1}{25} \].To simplify probabilities, you can reduce fractions like this one by dividing the numerator and denominator by their greatest common divisor. This final step of dividing ensures that our answer is in its simplest form, which makes it easier to interpret and understand. The result, \( \frac{1}{25} \), tells us that there is only one favorable arrangement out of 25 total possibilities.